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V^     ANALYTICAL  GEOMETRY. 


BY 


\j^  ALBERT    E.    CHURCH,    LL.D., 


I'UOl'liSSOn    OF  MATHEMATICS    IN    Tllli:     U.    S.     MILITARY    ACADEMY;     AUTHOR    OF 

ELEMENTS   OF   THE   DIFFERENTIAL   AND   INTEGRAL   CALCULUS,   KLE- 

MliNTa  OF  DESCRIPTIVE   GEOMETRY,    ETC 


THIRD   EDITION. 


A.  S.  BARlSfES  AND   COMPANY, 

NEW  YORK  AND  CHICAGO. 

18  71. 


C     C     C     -«. 


eoocATraiN  £iepr.  *      V   a\ 

Talnalile  Woris  liy  Leadii  Antliors  ^  ^  A 


\ 


HIGHER  MITHEMITICS. 


y'rof.  Jfathemalics  in  the   U.  6\  Jft'U/a/y  c4cadev/j>,   ITest  iPoint. 

CHURCH'S    -AJ^^AIJYTIC^L    GEOHNIiETPiY. 

CHXJRCH'S    CAL.CXJIL.TJS. 

CHUHCH'S    DESCRIPTIVE    G^EOMIETRY, 


Zate  !Pr6f.  Jfat/iemalics  in  the  University  of  Yirffi7tia, 
COURTE]?^^Y'S    CAnL.CXJX.XJS. 


Jyaie  tProf.  of  Mathematics  and  ^strojionty  in   Cotninbia  Colteffe. 
HOCKLEY'S    TRIG^O]?^OMEXI?,Y. 


f>of.  ofJVat.  <£'  Uxj).  Thilos.  in  the  U.  S.  Military  :>lcad.,  Vest  Toint. 

BARTLETT'S  sy:>ttiietic  nvEECHAisrics. 

SARTX.ETT'S    ANJ5LILYTICAIL.    JVIECIIANICS. 
I3ARTIL.ETT'S    -A.COXJSTICS    A:N"X)    OPTICS. 
BARTLETT'S    ASTROlsrOlMY. 


i)epartmenl  of  Mathematics ,  Cotumbia  Colter/e. 
Mli^THEM^^XICi^IL.    DICTIOIlSru^kRY . 


OHAI^LES    13A.VIES,    Li:^.X>., 

t^:tie  of  the  United  States  Mititary  :^cadcmy  and  of  Columbia  Collef/e. 

A^    COiyLnLiETE    COURSE    IN"    MIATHEIMA-TICS. 

See  A.  S.  Babkes  «fe  Co.'s  Descriptive  Catalogue. 


Entered,  according  to  Act  of  Congress,  in  the  year  1851,  by 

ALBERT    E.     CHURCH, 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States  i<  r  the  Southern  District  of 

New  York. 

C.  A.  G. 


J> 


J^ 


PREFACE 


No  bruriwti  of  pure  Mathematics  presents  more  to  interest 
dJid  improve  the  mind  of  the  mathematical  student,  than 
Analytical  Geometry.  Uniting  the  clearness  of  the  geome- 
trical reasoning,  with  the  brevity  and  generality  of  the  al- 
gebraic, it  not  only  satisfies  the  requirements  of  tho  closest 
reasoner,  but  gives  continued  and  increasing  pleasure,  by 
the  elegance  with  which  its  varied  resuks  are  deduced  and 
interpreted. 

In  preparing  this  treatise  the  Author  has  endeavoured 
to  preserve  the  true  spirit  of  Analysis,  as  developed  by  the 
celebrated  French  mathematician,  Biot,  in  his  admirable 
Avork  on  the  same  subject,  while  lie  has  made  such  chan- 
ges, both  in  the  arrangement  of  the  matter  and  the  methods 
of  demonstration,  as  he  believed  would  render  the  whole 
more  attractive,  and  easily  acquired  by  any  student  pos- 
sessing a  knowledge  of  the  elementary  principles  of  Alge- 
bra and  Geometry. 

In  discussing  the  Conic  sections  he  has  preferred  to  con- 
sider the  Parabola  first,  not  only  for  the  reason  that  the  pro- 
perties of  this  curve  are  more  simple  and  more  easily  de- 
duced than  those  of  the  others,  but  because,  by  this  course. 


f,  4  4-4  09 


IV  PREFACE. 

he  was  enabled  to  treat  of  the  Ellipse  and  Hyperbola  to- 
gether, thus  avoiding  much  of  the  repetition  of  words,  which 
necessarily  arises  from  their  separate  discussion. 

Although  the  treatise  has  been  prepared  with  special 
reference  to  the  wants  of  the  Author's  own  classes  at  the 
Military  Academy,  he  trusts  that  it  will  be  found  accepta- 
ble and  useful  to  all,  who  are  disposed  to  advance  in  the 
higher  branches  of  Analysis. 

Those  who  desire  to  make  the  subject  as  practical,  as 
may  be,  will  find  in  the  last  article  of  the  work  a  large  num- 
ber of  examples. 

U,  S.  Military  Academy^ 
West  Point,  N.  Y.,  July  1,  1851. 


CONTENTS 


PART   I. 

DETERMINATE  GEOMETRY. 

Vsge 

Definition  an  i  division  of  the  subject 1 

Mode  of  representing  geomstrical  magnitudes *3 

Linear  expressions  and  equations 3 

Construction  of  algebraic  expressions 4 

"  of  the  roots  of  equations  of  the  second  degree 8 

Determinate  problems 11 

General  rule  for  the  solution  of  determinate  problems 23 


PART   II. 

INDETERMINATE  GEOMETRY. 

Mode  of  representing  points  in  a  plane 24 

Definition  of  rectilineal  co-ordinates  and  co-ordinate  axes 25 

Equations  of  a  point 25 

Expressions  for  distance  between  two  points,  in  a  plane 20 

Definition  of  polar  co-ordinates  •. 27 

General  definition  of  co-ordinates 28 

Of  the  right  line  in  a  given  plane 28 

Manner  of  constructing  a  line  from  its  equation 31 

Definition  of  the  equation  of  a  line 3*2 

Every  equation  of  the  first  degree  between  two  variables  represents  a 
-ight  line 33 


VI  CONTENTS. 

Pa§6 

Manner  of  determining  the  intersection  of  lines 3G 

Angle  included  by  two  right  lines 37 

Conditions  that  two  right  lines  be  parallel  or  perpendicular 88 

Equation  of  a  right  line  passing  through  one  point 39 

"  "  "  two  points ..40 

Lvery  equation  between  two  variables,  the  equation  of  a  line 42 

Classification  of  lines 42 

General  equation  of  the  circle 43 

Equation  and  discussion  of  the  circle  referred  to  its  centre 44 

When  a  line  passes  through  the  origin  of  co-ordinates 46 

Definition  of  co-ordinate  planes,  &c 47 

Equations  of  a  point,  in  space 4S 

Expression  for  the  distance  between  two  points,  in  space 49 

Polar  co-ordinates,  in  space 50 

Equations  of  the  right  line,  in  space 51 

Intersection  of  lines,  in  space 54 

Angle  included  between  two  lines,  in  space 56 

Condition  that  two  lines,  in  space,  be  parallel  or  perpendicular 59 

Equations  of  a  right  line  passing  through  a  point,  in  ^ipace GO 

"  '•  "  two  points     "       61 

Equations  of  curves,  in  space 62 

Equation  of  a  plane 64 

Equations  of  the  traces  of  a  plane 66 

Every  equation  of  the  first  degree,  between  three  variables,  is  the  equation 

of  a  plane ^. 67 

Intersection  of  lines  and  planes 68 

Conditions  that  a  right  line  shall  be  perpendicular  to  a  plane 70 

Angle  between  a  right  line  and  plane 71 

Intersection  of  two  planes 72 

Conditions  that  two  planes  shall  be  parallel 73 

Angle  between  two  planes 74 

Condition  that  two  planes  shall  be  perpendicular 75 

Equation  of  a  plane  passing  through  one  point 77 

"  "  "  two  points 78 

Transformation  of  coordinates 78 

Eormulas  for  passing  from  one  system  to  another 81 

■'  "        to  a  system  of  polar  co-ordinates 84 

Polar  equation  of  the  circle 85 

Two  classes  of  propositions  in  transformation  of  co-ordinates 87 

Formulas  for  transformation  in  space 89 


CONTENTS.  yi 

Formulas  for  transformation  to  polar  cQ-ordinates,  in  space . . .' OC 

Of  the  cylinder 91 

General  equation  of  cylinder 92 

Of  the  cone 93 

General  equation  of  the  cone 94 

Equation  of  right  cone  with  a  circular  base 97 

[ntersection  of  riglit  cone  with  a  circular  base,  and  plane 98 

Classification  of  conic  sections 10',i 

Equation  of  the  parabola 103 

Definition  of  the  axis  of  the  parabola  and  discussion  of  equation 104 

Manner  of  constructing  the  parabola 105 

Definiiion  of  the  parabola ;  of  its  focus  and  other  modes  of  construction. 107 

Squares  of  the  ordinates  proportional  to  the  abscissas 108 

Equation  of  the  tangent  line  to  the  parabola 108 

Expression  for  the  subtangent 1 1 1 

Diflierent  methods  of  constructing  tangent  lines  to  the  parabola Ill 

Polar  line  to  the  parabola 114 

Properties  of  tangents  at  the  extremities  of  a  chord  passing  through  the 

focus 117 

Equation  of  a  normal  to  the  parabola 118 

Parabola  referred  to  oblique  axes 119 

Definition  of  a  diameter,  and  mode  of  constructing  it I2l 

Parameter  of  any  diameter 122 

Area  of  the  parabola 124 

Polar  equation  of  the  parabola 125 

Equation  of  the  ellipse  and  hyperbola 129 

Discussion  of  the  equation  of  the  ellipse  referred  to  its  centre  and  axes.132 

Equation  of  the  hyperbola  referred  to  its  centre  and  axes 133 

Equilateral  hyperbola 130 

Parameter  of  the  ellipse  and  hyperbola 138 

Foci  and  eccentricity  of  the  ellipse  and  hyperbola 139 

Definition  and  construction  of  an  ellipse 141 

"  "  of  an  hyperbola 142 

Property  of  the  foci  of  ellipse  and  hyperbola 145 

Equations  of  ellipse  and  hyperbola  referred  to  the  principal  vertex  . .  ..14G 

Relation  of  the  squares  of  the  ordinates 147 

Mode  of  constructing  ellipse  with  a  ruler 149 

Equation  of  tangent  to  the  ellipse 150 

Property  of  subtangent,  and  construction  of  tangent 152 

Equation  of  tangent  to  hyperbola 153 


Vjii  CONTENTS. 

Page 
Prcperties  of  tci'o  lines  drawn  from  the  foci  to  the  point  of  contact  of  a 

tangent  154 

Construction  of  tangent  lines 155 

Corresponding  properties  and  constructions  fo^  the  hyperbola J  5(i 

Equation  of  condition  for  supplementary  chords 157 

Properties  of  supplementary  chords,  and  construction  of  tangents 162 

Polar  line  of  the  ellipse 163 

"         of  the  circle  and  hyperbola 165 

Equation  of  a  normal  to  the  ellipse 166 

Area  of  the  ellipse 167 

Conjugate  diameters  of  the  ellipse  and  hyperbola 168 

Equation  of  condition  for  conjugate  diameters  in  the  ellipse 171 

"  "  "  "        in  the  hyperbola 173 

Parameter  of  any  diameter 175 

Construction  of  the  ellipse  and  hyperbola,  two  conjugate  diameters  be- 
ing given 176 

Construction  of  diameters  and  tangents 177 

Parallelogram  on  conjugate  diameters,  &c 182 

Equal  conjugate  diameters  of  the  ellipse 183 

The  asymptotes  of  the  hyperbola 181 

Equation  of  the  hyperbola  referred  to  its  centre  and  asymptotes 187 

Power  of  the  hyperbola 188 

Equation  of  tangent  referred  to  the  asymptotes 190 

Polar  equations  of  the  ellipse  and  hyperbola 192 

Discussion  of  the  general  equation  of  the  second  degree 199 

Equation  of  a  diameter 201 

Equation  of  second  degree  when  ^  =  0,  c  =  0 206 

Classification  of  the  conic  sections  represented  by  the  equation 207 

General  discussion  of  the  parabola 207 

Limits  of  the  parabola 208 

Particular  cases 209 

Practical  examples 210 

Construction  of  the  parabola  from  its  equation 213 

General  discussion  of  the  ellipse 213 

Limits  of  the  ellipse 215 

Particular  cases 215 

Practical  examples 216 

Construction  of  the  ellipse  from  its  equation  217 

General  discussion  of  the  hyperbola 219 

Particular  case 220 


CONTENTS.  LI 

Pag* 

Practical  examples 221 

Construction  of  the  hyperbola  from  its  equation 221 

Definition  and  discussion  of  centres  of  curves 222 

Application  to  lines  of  the  second  order 223 

Definition  and  discussion  of  diameters 225 

Of  loci 230 

Of  surfaces  of  revolution 238 

General  equation  of  surfaces  of  revolution 239 

"  "         when  axis  of  Z  is  the  axis  of  revolution 241 

Examples 241 

Classification  of  surfaces  of.  the  second  order 244 

Intersection  of  every  such  surface  is  a  line  of  the  second  order 245 

Every  system  of  parallel  chords  of  such  surface  may  be  bisected  by  a 

plane 246 

General  equation  of  surfaces  of  the  second  order 247 

discussion  of  such  equation 248 

Equations  of  the  three  classes 250 

Centres  of  surfaces  of  the  second  order 250 

Definition  of  diametral  and  principal  planes 251 

Of  the  ellipsoid 252 

Its  particular  cases  255 

Of  the  hyperboloid  of  two  nappes 256 

Its  particular  cases 259 

Of  the  hyperboloid  of  one  nappe 260 

Its  particular  cases 262 

Of  the  elliptical  paraboloid 263 

Its  particular  case 264 

Of  the  hyperbolic  paraboloid 265 

Intersection  of  surfaces  of  the  second  order  by  planes 266 

When  the  intersections  are  right  lines 268 

Descriptive  classification  of  surfaces  of  the  second  order 273 

Circular  sections  of  surfaces  of  the  second  order 273 

Subcontrary  sections  in  a  cone 276 

Intersection  of  surfaces  of  the  second  order 278 

Equation  of  tangent  planes  to  surfaces  of  the  second  order 280 

Line  of  contact  of  cone  and  surface  of  the  second  order 283 

Tangent  plane  passing  through  a  right  line 283 

Equations  of  normal  line  to  surfaces  of  the  second  order 284 

Practical  examples 385 


ANALYTICAL    GEOMETRY. 


PART     I 


DETERMINATE    GEOMETRY. 

1.  Geometry,  in  its  most  general  sense,  lias  for  its  object,  not 
only  the  measurement,  but  the  development  of  the  properties  and 
relations,  of  lines,  surfaces,  and  volumes. 

This  object  may  be  attained,  either  by  operating  directly  upon 
the  magnitudes  themselves ;  or,  by  representing  them  and  their 
parts,  by  algebraic  symbols,  and  operating  upon  these  representa- 
tives by  the  known  methods  of  Algebra,  thus  deducing  results  es- 
sentially the  same  as  those  which  would  be  obtained  by  the  direct 
method.  As  the  reasoning  employed  is  much  generalized,  and 
operations  are  much  abridged  by  the  application  of  Algebra,  the 
latter  method  evidently  possesses  many  advantages  over  the 
former. 

This  latter  method,  which  is  Analytical  Geometry,  may  be  de- 
fined to  be  :  That  branch  of  Mathematics,  in  which,  the  magni- 
tudes considered  are  represented  by  letters,  and  the  properties  and 
relations  of  these  onagnitudes  made  known  by  the  application  of  tJvs 
various  rules  of  Algebra. 

Analytical  Geometry  may  be  Determintte,  or  Indeterminate. 
2 


2  DEIERMINATE    GEOMETRY. 

De(efrMi?in(e,  when  it  lias  for  its  object  the  solution  of  determi- 
nate problems,  that  is,  of  problems,  in  which,  the  given  conditions 
limit  the  number,  and  aflford  the  means  of  deducing  the  values,  of 
the  required  parts. 

Indeterminate,  when  it  has  for  its  object  the  discussion  of  the 
general  properties  of  geometrical  magnitudes. 


2.  Geometrical  magnitudes  may  be  represented  algebraically, 
in  two  ways. 

First.  The  magnitudes  may  be  directly  represented  by  letters ; 
j[ a 3  as  the  line  AB,  given  absolutely,  may  be  re- 
presented by  the  symbol  a.  Likewise,  the 
square  AC,  may  be  represented  by  the  sym- 
bol A ;  or  better  by  the  symbol  a'^,  a  being 
the  representative  of  the  side  AB.  Also,  the 
rectangle  ABO'D'  may  be  represented  by  tho 
symbol  B  ;  or  by  the  product  ah,  a  and  6  be- 
ing the  representatives  of  the  sides  AB  and 
B  C ' ;  or,  better  still,  by  c^  c  representing 
the  side  of  a  square  equivalent  to  the  rectan- 
gle. In  the  same  way,  a  cube  would  be  re- 
presented by  a^  a  being  the  representative  of 
and  a  rectangular  parallelopipcdon  by  ahc  or 


J?' 


B 

or* 
ah  Of  **2 


one  of  the  edjres 


hy(P, 

And  in  general,  we  thus  represent  a  definite  portion  of  a  hne, 
whether  straight  or  curved,  by  a  single  letter  or  expression  of  the 
first  degree  ;  a  surfjice  by  the  product  of  two  letters  or  an  expres- 
sion of  the  second  degree  ;  and  a  volume  by  an  expression  of  the 
third  degree. 

Second.  Instead  of  representing  the  magnitude  directly,  the  al' 
gebraic  symbol  may  represent  the  number  of  times,  that  a  given 
or  assumed  unit  of  measure  is  contained  in  the  magnitude  ;  as, 
for  the  line  AB,  a  may  represent  the  number  of  times  that  a 


DETERMINATE    GEOMETRY.  8 

known  unit  of  length  is  contained  in  it ;  and  a^  and  ah  or  c",  the 
number  of  times  that  a  square  whose  side  is  the  unit  of  length,  is 
contained  in*  the  given  square  or  rectangle  ;  and  a?  and  ahc^  the 
number  of  cubic  units  contained  in  the  given  cube  or  parallelopi- 
pedon. 

Since,  in  this  case,  the  algebraic  symbols  represent  abstract 
numbers,  any  algebraic  expression,  thus  composed,  is  called  an 
abstract  expression  or  equation,  to  distinguish  it  from  one  in  which 
the  direct  representatives  of  the  magnitudes  enter.  Since  a  line 
is  always  represented  by  an  algebraic  expression  of  the  first  degree, 
such  expression  is  called  linear.  Also,  a  linear  equation  is  an 
equation  of  the  first  degree. 


3.  From  what  precedes,  it  is  evident,  that  any  abstract  expres- 
sion may  be  changed  into  one  in  which  the  direct  representatives 
of  the  magnitudes  enter,  Z»y  substituting,  for  the  representative  of 
each  abstract  number,  the  representative  of  the  magnitude  divided 
by  the  representative  of  the  unit  of  measure.     Thus  in  the  expression, 

aj  =  a  +  6, 

X,  a  and  b  representing  numbers  ;  if  we  substitute  for  them,  their 

equals  — ,    — ,    — ,  X,  A  and  13  being  the  direct  representatives 

L  t  L 

of  the  magnitudes,  and  I  that  of  the  unit  of  measure,  w^e  have 

^  =  A.l        or         X  =  A  +  B. 
I  I    ^    I 

In  the  same  way,  the  abstract  expression 

X  =^  ab  -{-  c, 

may  be  changed  into  the  corresponding  one, 

^  =  A  ^  +  ^        or         XZ  =  AB  +  C/. 
/  I     I    ^    I 


4  DETERMINATE    GEOMETRY. 

It  should  bo  remarked,  that  every  expression  of  this  kind  must 
be  homogeneous,  else  we  should  have  magnitudes  of  ditiereut 
kinds  added  or  subtracted  or  equal,  which  can  not  be. 

4.  After  having  deduced  a  result,  by  the  application  of  algebra 
to  a  geometrical  proposition,  it  will  be  necessary  to  explain  this 
result  geometrically,  that  is,  to  draw  a  geometrical  figure,  in  lohick 
shall  be  found  each  of  the  lines  represented  in  the  algehraic  expres- 
sion^ and  the  geometrical  relation  between  these  lines  shall  be  the 
same  as  that  indicated  in  the  expression.  lLh.\%\?>CQ\\Q,(\.  constructing 
the  expression. 

Examples. 

1.  Let  X  —  a  -^  b. 

If  a  and  b  are  the  direct  representatives  of  right  lines,  x  will  be 
the  representative  of  their  sum.  To  construct  it,  take  the  line  re- 
presented by  a,  in  the  dividers,  and  from  any  point  A,  on  the 

^^___—      indefinite  line  X'X  as  a  point  of 

"^  beginning,  or  origin,  lay  off  AB 

equal  to  this  distance,  then  from  B  lay  off  BC  equal  to  the  line 
representee!  by  b,  the  line  AC  =  AB  +  BC  will  evidently  be 
represented  by  x. 

Or  if  a  and  b  represent  numbers,  lay  off  from  A,  a  times  the 
unit  of  length,  then  from  B,  b  times  the  same  unit,  and,  as  before, 
AC  will  be  the  line  represented  by  x. 

2.  Let  X  =  a  —  b. 

From  A  lay  off  AB  =  a,  then  from  B  lay  off,  towards  A,  thp 

___^^_     distance    BC  =  J ; 

ic  6"  >L         cr      -zi      X  AC  =  AB  -  BC 

will  be  the  line  represented  by  x. 


DETERMINATE    GEOMETRT.  6 

If  a  =  h,  X  will  be  equal  to  0,  the  point  C  will  evidently 
fall  on  A,  and  there  will  be  no»line. 

If  Z>  >  a,  X  will  be  essentially  negative,  the  point  C  will 
fall  on  the  left  of  A,  as  at  C,  and  AC,  laid  off  from  A  to  the  left, 
will  be  represented  by  x.  Thus,  we  see  an  illustration  of  the 
principle  taught  in  Trigonometry,  that  if  lines  having 'the  positive 
siojn  are  estimated  or  laid  off  in  one  direction,  those  having:  the 
negative  sign  must  be  estimated  in  a  contrary  direction. 


3.  Let-  x=.'±. 

c 

In  this  case  a;  is  a  fourth  proportional  to  c,  a  and  i,  and  is  thus 
constructed.  Draw  any  two  right  lines 
making  an  angle  ;  on  one,  from  their 
point  of  intersection,  as  an  origin,  lay  off 
the  distances  AC  =  c  and  AB  =  a ;  JET  G  3" 
on  the  second,  lay  off  AD  =  h  ;  join  the  points  G  and  D,  and 
through  B  draw  BX  parallel  to  CD  ;  AX  will  be  the  line  repre- 
sented by  x.     For,  we  have 

AC  :  AB  :  :  AD  :  AX        or        c  :  a  : :  J  :  AX 


whence 


AX  =  f?  =  z. 


4.  Let  X  =z  — 

de 


This  may  be  put  under  the  form 

ah  c 

a;  =  -  X  - 

d  § 


DETERMINATE    GEOMETRY. 

Place    _-  =  <7,  and  construct  g  as  above,  then  we  have 
a 


x=?l, 


which  may  be  constructed  in  the  same  way ;  and  so  with  any  ex- 
pression, in  which  the  number  of  factors  in  the  numerator  is  one 
greater  than  in  the  denominator. 


5.  Let  X  =   Voft  or  «'  =  ah. 

In  this  case,  a:  is  a  mean  proportional  between  a  and  b.  To 
construct  it :  On  any  right  line,  lay  off  AB  =  a  ;  from  B  lay 
off  BC  =  6  ;  on  the  sum,  AC,  describe  a 
semi-circle,  and  at  the  point  B  erect  BX 
perpendicular  to  AC.  The  jJart  BX,  in- 
cluded between  the  diameter  and  circum- 
ference, will  be  the  line  represented  by  x. 
For  from  a  known  property  of  the  circle,  we  have 


BX'  =  AB  X  BC        or        BX  =   V^  = 


X, 


-    T    .  ^  /  abc  ^  /ab 

C.  Let  ^  =  Vt-  =   Vy   ""■ 

Place    —  =:  g  and  construct  it  as  in  example  3,  then  we  have 
d 

X  =   ^/gCy 

which  may  be  constructed  as  above. 


7.  Let        X  =   Va'  +  &*        or        a;«  =  a«  -f  J«, 


DETERMINATE    GEOMETRT. 


In  tliis  case,  x  is  the  hypotlienuse  of  a  right  angled  triangle,  the 
two  sides  of  which  are  a  and  h.  Therefore,  draw  two  lines  form- 
ing, with  each  other,  a  right  angle  :  From 
the  vertex.  A,  on  one,  lay  oflf  AB  =  a ; 
on  the  other,  lay  off  AC  =  h ;  join  B 
and  C,  the  line  BC  will  be  represented  by 
X.     For  we  have 


BO^  =  AB'  +  AC' 


or 


BC  =   Va^  +  62  = 


8.  Let 


=   Va^  —   6*. 


From  A,  in  the  last  figure,  lay  off  AC  =  h  ;  then  from  C  as  a 
centre,  and  with  CB  =-  a  as  a  radius,  describe  an  arc  cutting  AB 
in  B ;  the  distance,  AB,  will  be  represented  by  x.     For 

AB  =   Vbc'  -   AC'  =    Va'  -   h^  =  x. 


9.  Let 


Va^  +  62  __" 


Place  a^  +  6*  =  g^^  and  construct  g  as  in  exarcple  7  ;  then 
we  have 

which  may  be  constructed  as  above. 


10.  Let 

X  =   Va^  +  CLC, 

11.  Let 

ahc  -\-  g^d 

/  •>: 

12.  Let 


8 


DETERMINATE    GEOMETRY, 


6.  Let  US  now  construct  tlie  roots  of  tlie  four  forms  of  equations 
of  the  second  degree.     The  firsts 


x^  +   2a^  =    y^, 


gives  the  roots 


—  a  +    VZ»2  +  a^, 


—  «  —    s/W"  + 


From  any  point,  as  A,  lay  off  AB  :^  &  ;  at  B,  erect  the  per- 
pendicular    BC  =  a,     then  as   in 
^      example  7 

\c  

AC  =    VZ/^  +  a2. 

Now  from  C,  as  an  origin,  lay  off 


=  AD 


CD  =  a,     then 

AC  -  CD  =   Vi^  +  a?-  - 

will  be  represented  by  the  first  value  of  x. 

From  E,  layoff    EC  =  a,     also     CA  =   Vb^  +  a« ;   then 

-  EC  -  CA  =   -  a  -    Vb»  +  a^  =   -  EA 

will  be  represented  by  the  second  value  of  x. 

The  given  equation  may  be  put  under  the  form 

x{x  +  2a)  =   l^, 

from  which  we  see  that  b  is  a  mean  proportional  between  x  and 
X  +  2a,  and  this  relation  -will  be  satisfied  by  either  of  the  above 
lines  AD  or  —  EA.     First,  by  substituting  AD  for  a;,  we  have 

AD(AD  +  2a)  =  V^        or        AD(AD  +  DE)  =  AB', 
as  it  should  be,  since  AB  is  a  tangent,     AD  +  DE   —  AE,     a 


DETERMINATE    GKOMETRY.  9 

secant,  and  AD  its  external  part.     Second,  bv  substituting  —  EA 
for  X 

—  EA(-  EA  +  2a)  =  62        ^^        EA  x    AD  =  AB*- 

The  second^ 

x^  —   2ax  =  h% 
gives  the  roots 


X  —  a  +   Vb*  +  a^  X  =  a  —   Vb^  +  a\ 

Construct  as  before,  AC  —   Vl^  +  a'^ ;     tlien  from  C  lay  off 
CE  =  a,  and 

AC  +  CE  =   Vh-'  +  a^  +  «  =  AE, 

will  be  represented  hy  the  first  value  of  x. 

From  D,  lay  off    DC  =  a ;     then  from  C  in  a  contrary  di 
rection  lay  off    CA  =    Vh"^  +  a^  and 


k 


DC  -  CA  =  (z  -   V62  4-  a2  =   —  DA 

will  be  represented  by  the  second  value  of  x. 
The  given  equation  may  be  put  under  the  form 

x{x  —   2a)  =    6^ 

which  will  evidently  be  verified  by  the  substitution  of  either  AE 
or  -  AD. 

It  should  be  observed  that  the  values,  just  constructed,  are  the 
same  as  those  for  the  first  form,  with  their  signs  changed.  This 
should  be  so,  since  the  first  form  will  become  the  second  by 
changing  x  into  —  x. 

The  third 

x^  +  2ax  =z   ^  69, 


10 

gives  the  roots 


DETERMIXATE    GEOMETRY. 


X  = 


V^F^b^ 


From  A  as  an  origin,  on  the  line  AA'  lay  off  the  distance 
—  AD  =   —  a ;     at  D  erect  the   perpendicular     DC  =   h  j 
C  from  C  as  a  centre,  with  CB  =  a, 

as  a  radius,  describe  the  arc  BB' 
cutting  the  line  AA'  in  B  and 
V    1       B' ;  join  these  points  with  C  and 
,     we  shall  have  DB  =  Va'*  —  6% 


—  AD  +  DB=—  a+   Va^  -  h^  =  -  AB 

-  AD  —  DB  =   -  a  -   Va2  -  6=*  =   -  AB' 

will  be  the  lines  represented  by  the  values  of  x. 
The  fourth, 


x^  —  2aa;  =  —  Z>*, 


gives  the  roots 


a  —   Va^  —  h\ 


From  A',  as  an  origin,  lay  off    A'D  =  a,    and  make  the  same 
construction  as  for  the  third  form.    We  thus  have 


A'D  +  DB  =  a  +   Va2  -  6«  =  A'B 
A'D  -  DB'  =  a  -   Va2  -  h^  =  A'B' 

for  the  lines  represented  by  the  values  of  x. 

If  a  =  b,  both  values  of  x  reduce  to  a  =  A'D.  In  this 
case,  the  circle  does  not  cut  the  line  AA',  but  touches  it  at  the 
point  D,  and  the  distances  BD  and  B'D  become  0.     The  same 


DETERMINATE    GEOMETRY. 


11 


8U23position,   in   the   third  form,   reduces   both   values  of  x   to 
—  AD. 

If  a  <C^  h,  the  values  of  x  become  imaginary  in  both  forms  ; 
the  circle  neither  cuts  nor  touches  the  line  AA',  and  the  imagina- 
ry roots  admit  of  no  construction. 

DETERMINATE    PROBLEMS. 


6.  A  thorough  knowledge  of  the  preceding  principles,  will  ren- 
der the  solution  of  all  determinate  problems  simple  and  easy. 

Problem  1.  In  a  given  triangle^  to  inscribe  a  square. 

Let  ABC  be  the  triangle.  Represent  its  base,  AB,  by  6,  and 
its  altitude  CG  by  h.  Suppose  the  problem  to 
be  solved,  and  that  ODEF  is  the  required 
square,  its  unknown  side  DE  =  EF  being 
represented  by  x.  Since  the  side  DE  is  parallel 
to  AB,  we  must  have 


AB  :  DE 

whence 


CG  :  CH 


hx  =  bh  —   bx 


or 


and 


b  :  X  : :  h  :  h  ~ 


bh 


b  +  h 


hence  a:  is  a  fourth  proportional  to  5  +  ^,  b  and  h,  and  may 
be  constructed  as  in  example  3,  Art.  (4).  Or  better  thus  :  Pro* 
duce  the  base  AB  until  BL  =  h  ; 
at  B  and  L  erect  the  perpendiculars 
BN  and  LM ;  make  LM  =  h  and 
join  M  and  A  ;  the  part  BN  cut  off 
on  the  first  perpendicular  will  be 
represented  by  x. 


For,  since  BN  is  parallel  to  LM,  we  have 


AL  :  AB  :  :  LM  :  BN 


or 


5  +  A  :  6  : :  A  :  BN 


whence 


12 


DETERMINATE    GEOMETRY. 


BX   = 


hh 


Therefore,  througli  N  draw  ND  parallel  to  AB ;  let  fall  tho 
perpendiculars  EF  and  DO,  and  the  square  ODEF  will  be  the  re- 
quired square. 

The  value  of  rr,  and  the  construction  of  BN,  will  evidently  be 
the  same  for  all  triangles  having  the  same  base  and  equal  alti- 
tudes.    K  all  the  angles  of  the  triangle  are  acute,  the  square  will 

lie  wholly  within  the  triangle  as 
in  the  above  figure.  If  there  is 
one  right  angle,  two  sides  of  the 
square  will  lie  upon  the  sides  of  the 
triande  as  AD'E'F'.  If  there  is 
JL       JF  o^   B±^"  J^       one  obtuse  angle,  part  of  the  square 

will  He  within  and  part  without  the  triangle,  as  OT)"E"F''. 


V.  Problem  2.  In  a  given  triangle,  to  inscribe  a  rectangle^  the 
ratio  of  whose  adjacent  sides  is  known. 

Let  ABC  be  the  triangle.  Let  AB  =  &  and  CG  =  7i, 
and  let  the  known  ratio  of  the  sides  of  the  rect- 
angle be  denoted  by  r.  Suppose  the  problem 
to  be  solved,  and  that  ODEF  is  the  required 
rectangle.  Denote  the  unknown  side  DE,  by  y, 
and  DO  by  x  ;  then  by  the  given  condition, 


0      G  F  3 
we  have 


:i-  =  r. 


.(1). 


Since  DE  is  parallel  to  AB,  we  have 

AB  :  DE  :  :  CG  :  CH        or        I  \  y  \\h  \    h  ^'  x 

whence 

hy  =  Ih  —  bx. 

From  this,  by  substituting  the   value     y  =  rx,     taken  from 


DETERMINATE    GEOMETRY. 


13 


equation  (1),  we  deduce 

rhx  =^  bh  —  bx 


or 


hh 


b  +  rh 


To  const*:uct  this  value  of  x ;  produce  tlie  base  AB  until 
BL  =  rh ;  through  L  draw 
LM  parallel  to  BC  until  it 
meets  CM  parallel  to  AB,  in 
M ;  join  M  and  A ;  at  the 
point  E,  let  fall  EF  perpendic- 
ular to  AB,  it  will  be  the  re- 
quired hne.  For,  since  the  triangles  AEB  and  AML  are  similar, 
their  bases  will  be  to  each  other  as  their  altitudes,  and  we  shall  have 

AL  :  AB  :  :  MP  :  EF        or        b  +  rh  :  b  \:  h  :  EF 

whence 


EF  = 


bh 


b  +  rh 


z=    X 


Therefore,  through  E  draw  ED  parallel  to  AB,  and  let  fall  tha 
perpendiculars  EF  and  DO  ;  ODEF  will  be  the  required  rectangle. 
If   r  =  1,    the  sides  are  equal,  the  rectangle  becomes  a  square, 
and  we  have  th*e  same  value  for  EF  as  in  the  preceding  article. 


8.     Problem  3.  To  draw  a  straight  line  tangent  to  two  given 
circles. 

Since  the  two  circles  are  given,  both  in  extent  and  position,  we 
know  their  radii  and  the  distance  between  their  centres. 

Let  us  denote  the  radius,  CM,  of  the  first  circle  by  r,  that  of  the 
second,  CM',  by  r',  and  the 
distance  between  their  cen- 
tres, CC,  by  a,  and  suppose 
that  MM'  is  the  required 
tangent  and  denote  the  dis- 
tance CT  by  X, 


u 


DETERMINATE    GEOMETRY. 


There  are  two  cases  : 

First ;  when  the  tangent  does  not  ji>ass  between  the  circles. 

Since  the  radii  drawn  to  the  points  of  contact,  M  and  M',  must 
be  perpendicular  to  the  tangent,  we  have  CM  parallel  to  CM',  and 
hence  the  proportion 


CM  :  OW  : :  CT  :  C'T 
whence 


or 


r'  w  X  \  X  —  a, 


r'x  =  rx  —  ra 


and 


To  construct  this  value  of  x :  Through  the  centres  C  and  C,  draw 

any  two  parallel  radii 
^  CN  and  C'W,  on  the 

same  side  of  CC ;  join 
their  extremities  by  the 
line  NN'  and  produce 
it  until  it  meets  CC  in 
T;    CT  will  be  the  line 

represented  by  x.    For,  draw  "N'O  parallel  to  CC,  we  then  have 

NO  :  NC  :  :  ON'  :  CT        or        r  —  r'  i  r  :  :  a  :  CT 

whence 

CT  =  -^_  =  X. 
r  -  r" 

Therefore,  through  the  point  T,  draw  TM  tangent  to  one  of  the 
circles,  it  will  be  tangent  to  the  other. 

If  r  >  r',  the  value  of  x  is  positive,  and  the  point,  T,  will  be  on 
the  right  of  C. 

If    ?•  =  r',    the  two  circles  are  equal,  tlie  value  of  x  reduces  to 


DETERMINATE    GEOMETRY. 


15 


the  point  T  is  at  an  infinite  distance,  and  the  tangent  is  parallel  to 
CC. 

If  r  <  /,  the  value  of  x  is  negative,  and  the  point  T  is  on 
the  left  of  C. 

If  r  =:  0,  X  will  be  0,  the  first  circle  becomes  a  point,  and 
the  tangent  is  drawn  from  this  point  to  the  second  circle. 

If  r'  =  0,  X  will  reduce  to  a,  the  second  circle  becomes  a 
point,  and  the  tangent  is  drawn  from  this  point  to  the  first  circle. 


If    r  =   0     and 


0,     the  value  of  x  reduces   to   — , 


an  indeterminate  quantity^  each  circle  becomes  a  point,  and  the 
tangent  coincides  with  CC. 
'     Second  ;  when  the  tangent  passes  between  the  circles. 

In  this  case  as  in  the  other, 
the    lines   CM  and    CM'    are 

parallel,    hence,    the    triangles     j  ^^ ] n^  | ^^/ 

MCT  and  M'CT    are   similar, 
and  we  have  the  proportion 

CM  :  CM'  :  :  CT  :  CT,        or        r  :  r'  :  :  x  :  a 

whence 

r'x  =  ar  —  rx         and  x  = 


r  -{•  r' 


To  construct  this  :  Through  C  and  C  draw  any  two  parallel 
radii,  on  different  sides  of  CC  ; 
join  their  extremities  by  the  y^^  ^VM" 
line  NN' ;  CT  will  be  the  line 
represented  by  x.  For,  through 
N',  draw  N'O  parallel  to  CC, 
then  we  have  the  proportion 


NO  :  NC  : :  ON'  :  CT,        or        r  +  r'  :  r  : :  a  :  CT, 


16 

whence 


DETERMINATE    GEOMETRY. 


CT 


=  ar. 


r  +  /' 

The  value  of  x  is  positive  for  all  values  of  r  and  r' ;  reduces  to 
wlien    r  =  r' ;     to  0  when    r  =  0  ;  to  a  when    r'  =  0, 

and  to    — ,    when  r  and  r'  are  both  equal  to  0. 
0 


9.  Problem  4.  To  construct  a  rectangle^  Jcriowing  its  area  and 
the  difference  between  its  adjacent  sides. 

Let  a^  denote  the  given  area,  Art.  (2),  and  d  the  difference  be- 
tween the  sides.  Let  x  denote  the  least  side,  then  x  -\-  d  will 
denote  the  greatest,  and  since  the  rectangle  of  these  two  sides  must 
equal  the  given  area,  we  have 


X  {x  •\'  d)  =  a^         or         x^  +  dx 


whence 


o 


\/-  +  't 


If  we  take  the  first  value 


--l+V^'  +  f 

and  add  d  to  it,  we  have  for  the  greatest  side 

d 


X  -\-  d 


+ 


\/-  +  ? 


To  construct  these  values  :  Make  AB  =  a  ;  at  B,  erect  the 
perpendicular  BC  =  — ,  we  shall 
have.  Example  7,  Art.  (4), 

AC  =  \/a«  +  ^. 
^  4 


DETERMINATE    GEOMETRY.  17 


From   AC,    take     CD  =  — ,     and  wo  have 


AD  =   —  —  _{-\/^'  +  —  =  ^   =  the  least  side. 


To  AC,  add  CE  =  jl,        and  we  have 


AE  =  —  ^  v/  a^   -f-  ■—  X  -\-  d  =   the  greatest  side, 

and  the  rectangle     AE  x    AD  =  a^     will  be  the  required  rect- 
angle. 

If  we  take  the  second  value 


=  -  -T  -  V  «'  + 


2  ^4 

and  add  d  to  it,  we  have  for  the  greatest  side 


X  +  d  =  A.  ^  \/a^   +  —. 
2  V  '     4 

By  examining  these  values,  we  see,  that  the  expression  for  the 
least  side,  taken  with  a  negative  sign,  is  the  same  as  that  for  the 
greatest  side,  in  the  first  case.  Also,  that  the  expression  for  the 
greatest  side,  taken  with  a  negative  sign,  is  the  same  as  that  for 
the  least  side,  in  the  first  case.     Therefore  we  have,  in  tliis  case, 

-—  AE, '  for  the  least  side, 

—   AD,    for  the  greatest  side, 

the    product    of    which    is    evidently    positive— and    equal    to 
AB*  =  a\ 

It  should  be  observed,  that  it  is  c  nly  in  an  algebraic  sense, 
that  —  AE  is  less  than  —  AD,  its  numerical  value  being 
evidently  the  greatest. 


18  nCTKr.MINATE    GEOMETRY. 

It  is  also  evident,  lli.it  the  two  rectangles  thus  determined  are 
absolutely  equal,  or  that  in  reality  there  is  but  one  rectangle  ■which 
will  fulfil  the  required  conditions.  Why  then,  it  may  be  asked, 
do  these  conditions  lead  to  an  equation  of  the  second  degree  ?  To 
this  it  may  be  answered,  that  in  Algebra,  properly  applied,  not 
only  are  problems  solved  in  their  most  general  sense,  every  pos- 
sible solution  being  given  by  the  equation,  which  is  the  algebraic 
statement  of  the  problem  ;  but  also,  whenever  the  conditions  of  a 
problem,  expressed  in  two  independent  ways,  give  rise  to  the  same 
equation,  this  equation  must  give  an  answer  corresponding  to  each 
mode  of  expressing  the  conditions ;  that  is,  must  be  of  the  second 
degree,  and  it  will  thus  be  impossible  to  arrive  at  one  solution  dis- 
connected from  the  other. 

Thus,  in  the  above  problem,  should  we  represent  the  greatest 
side  by  —  .r,  the  least  would  be  represented  by  —  a;  —  rf, 
and  their  product  give 

—  X  {—  X  —  d)  =  a^         or         x^  •]-  dx  =  a«, 

the  same  equation  found  before ;  hence,  tHs  equation  ought  not 
only  to  give  the  least  side,  as  at  first  proposed,  but  also  another 
value  of  ar,  which  taken  with  a  negative  sign,  will  represent  the 
2freatest  side  of  the  rectangle. 


10.  Problem  5.  To  divide  a  given  straight  line  into  extreme 
and  mean  ratio. 

Let  AB  =  a  be  the  given  line.  It  is  to  be  divided  into  two 
parts,  such,  that  the  greater  shall  be  a  mean  proportional  between 

the   whole    line   and  less 


part.  Denote  the  greater 
part  by  a*,  then  a  —  x 
will  denote  the  less  part, 
and  the  condition  will  give 


n 


=  a  (rt  —  a:),         or         r*  +  ax  =  a'*, 


^  V    *ru 


DETEKMINAlti    GKOMETRY.  19 


whence 


+ 


Va^   +    .J,        .=    -|_v'a^-.| 


wliicb  may  be  constructed  precisely  as  in  the  preceding  problem, 
the  first  being  AD  and  the  second  —  AD'.  AVith  A  as  a 
centre,  and  AD  as  a  radius,  describe  the  arc  DF,  the  line  will  be 
divided  in  the  required  ratio  at  F,  AF  being  the  greater  part. 

The  second  value  of  a;  =  —  AD'  is  numerically  greater 
than  AB.  It  can  then  form  no  part  of  it,  and  can  not  be  an 
answer  to  the  proposed  question.  But  if  we  substitute  it  for  x  in 
tbe  first  equation,  we  have 


(-  AD')*  =  a  [a  -  (~  AD')]      or      AD'    =  a  (a  +  AD') 

that  is,  AD'  is  a  mean  proportional  between  AB  and  AB  -j-  AD'. 
Since  this  second  value  of  x  is  negative,  we  lay  it  off  to  the  left  of 
A,  and  thus  construct  the  point  F',  the  distance  from  which  to  A, 
is  a  mean  proportional  between  its  distance  from  B  and  the  length 
of  the  given  hne. 

^loreover,  we  see  that  the  second,  as  well  as  the  first  value  of 
.r,  is  a  solution  of  the  more  general  proposition,  "  Two  points,  A 
and  B,  being  given,  to  find,  on  the  indefinite  line  which  joins 
them,  a  third  point,  the  distance  from  which  to  the  first  shall  be  a 
mean  proportional  between  its  distance  from  the  second  and  the 
distance  between  the  two."  To  this  proposition  there  are  evi- 
dently two  solutions,  F  on  the  right  of  A  being  one  of  the  points, 
and  F'  on  its  left,  the  other.  Thus,  the  problem  at  first  proposed 
being  a  particular  case  of  a  more  general  one,  its  solution,  in 
accordance  with  the  principle  laid  down  in  the  jDreceding  article, 
must  necessarily  draw  with  it  that  of  the  other  case,  thus  giving 
rise  to  an  equation  of  the  second  degree. 


11.     Problem  6.   Through  a  given  point  without  a  given  angle^ 


20  DETERMINATE    GEOMETRY, 

to  draio  a  straight  line,  cutting  the  sides  of  the  angle,  so  that  the 
sum  of  the  distances  from  the  points  of  intersection  to  the  vertex^ 
shall  be  equal  to  a  given  line. 

Let  YAX  be  the  given  angle,  and  M  the  given  point.    Produce 
AX  to  the  left,  and  let  the  two  distances  MP'  and  MP  be  repre- 
-j^  sented  by  a  and  b.    Denote  the  given 

line  by  c.     Suppose  MN  to  be  the  re- 
^/"^  quired   line,  and  denote  the  two  un- 

known distances,  AN  by  x  and  AO 


JC-j2r. 


^     ^  "^      by   y.     Then  from  the  condition  of 

the  problem,  we  have 

AN  -f   AO  =  c        or        a:  +  y  =  c... (1). 

But  since  MP  is  parallel  to  AO,  we  have 

PN  :  AN  :  :  PM  :  AO,        or        a  ■{-  x  :  x  :  :  b  :  y; 

whence 

7j  {a  +    x)  -  hx (2). 

Substituting  the  value  of  x,  deduced  from  equation  (1),  we  have 
3/  (a  +  c  —  y)   =  6  (c  -   ?/) 

y  {a   +   c   -\-   b   —   tj)   —   be. 

This  being  an  equation  of  the  second  degree,  its  roots  may  be 
deduced  and  constructed  as  in  Art.  (5)»  But  by  examining  it,  in 
its  present  form,  we  see  that  Vbc  is  the  ordinate  of  a  circle 
whose  diameter  is  a  ■\-  c  -f-  b,  and  the  corresponding  seg- 
ments of  the  diameter,  y  and  a  +  c  +  ^  —  ?A  which  leads 
to  a  simple  construction  of  the  value  of  y.  Thus  :  From  P', 
lay  off,  on  AY,  P'B  =  c ;  also  BC  =  a ;  on  AB  describe 
the  semicircle  ALB  ;  at  P'  erect  the  ordinate  P'L,  it  will  be 


DETERMIXATK    GEOMETRY. 


21 


rejH'eseiited  by  V  6c,  Example 
5,  Art.  (4).  Through  L  draw 
LC  parallel  to  AY ;  then 
through  A  and  C  draw  the  per- 
pendicular AA'  and  CC  ;  A'C 
will  be  equal  to  a  +  c  +  Z> ; 
on  this  line  describe  th^  semi- 
circle CO  A' ;  the  distance  from 
the  point  0,   in  which  it  cuts 

AY,  to  A  will  be  represented        P         /A.  A"      X 

by  y.    For     00'  =  P'L,    and  the  segment    A'O'  =   AO, 
flls  the  required  condition. 


ful- 


12.  Problem  Y.  Through  a  given  point,  lolthout  a  given  angle, 
to  draw  a  straight  line,  so  as  to  cut  off  a  given  area. 

Let  the  given  point  and  angle  be,  as  in  the  first  figure  of  the 
preceding  article.  Let  7i'  represent  the  given  area,  and  /3  the 
given  angle.  The  expression  for  the  measure  of  the  required  tri- 
angle will  be  "lOT  x  AN.  From  the  right  angled  triangle 
OAT,  we  have 

OT  =  OA  sin  YAX  =  y  sin  /S  ; 

hence  the  area  will  be  expressed  by 

fry  sin /3. 

Substituting  the  value  of  x,  taken  from  equation  (2),  of  the  pre- 
ceding article,  and  placing  the  result  equal  to  A',  we  have 


ay 


2  h  —  y 


y  sm 


which,  by  reduction,  becomes 


y«   + 


h\ 


2h% 


Ih^y 

a  sin  j5         a  sin  /3 


22 


DETERMINATE    GEOMETRY. 


Solving  this  equation,  we  obtain 


y  = 


h^ 


h' 


a  sm 


+ 


2h% 


a^  sin^  /3  a  sin  jS 


To  construct  these  values  :  Through  A  draw  AA'  perpendicu- 
lar to  AY ;  then  since  the  angle 
A'PA  =  ^,     we  shall  have 

AA'  =  AP  sin  ^   =  a  sin  /S. 

Upon  AY,  in  a  negative  direction, 
lay  off  A13  —  h ;  join  A'  and  B, 
and  at  B  erect  BC  perpendicular  to 


A'B,  then 


or 


AC 


AB    =  AA'  X    AC 


a  sin  |8 


Since  this  expression  is  negative  in  the  above  values  of  y,  we  lay 
it  off  from  A  to  D.  The  radical  part  of  these  values  may  be  put 
under  the  form 


V     26    +    — T— 3)  —r 
^  \  a  sm  py  a  SI] 


sin  /3 

To  construct  it,  we  lay  off    P'S  ==  i  ;     on  SD  describe   a 
semi-c.rcle,  the  chord  DE  will  be  the  value  of  the  radical,  for 


AD  = 


h^ 


a  sin  /3  ' 


DS  =  26  +  AD, 


and  DE  is  a  mean  proportional  between  them.  From  D  lay  off 
DO  :-  DE,  and  AO  will  be  represented  by  the  first  vaUie  of  y. 
From  D  lay  off  DO'  =  DE,  and  AO'  will  be  represented  by 
tlie  second  value  of  y.  Through  the  points  O  and  C,  draw  MO 
and  MO',  and  either  triangle  cut  off  will  fulfil  the  condition  of  the 
problem. 


DETERMINATE    GEOMETRY.  23 

10.  By  an  examination  of  the  manner  in  which  the  preceding 
problems  have  been  solved,  we  may  derive  the  following  general 
rule  for  solving  determinate  problems. 

Conceive  the  iwoblcm  to  he  solved  geometrically,  and  draw  a 
figure  containing  the  given  and  required  parts,  and  such  other 
lines  as  may  he  necessary  to  show  the  relation  hetween  them.  Re- 
■present  the  known  lines  hy  the  first,  and  the  unknown  hy  the  last 
letters  of  the  alphahet.  Consider  the  geometrical  relations  existing 
hetween  these  lines,  and  express  them  hy  equations,  taking  care  to 
deduce  as  many  equations  as  there  are  unknown  quantities.  Solve 
these  equations  and  construct  upon  a  single  figure  the  values  thus 
deduced. 

By  an  application  of  this  rule  the  following  problems  are  readily 
solved. 

8.  Through  a  given  point  without  the  circumference  of  a  circle-, 
to  draw  a  straight  line  intersecting  it,  so  that  the  chord  included 
within,  shall  be  equal  to  a  given  line.   ■ 

r    9.     To  draw  a  line  parallel  to  the  base  of  a  triangle,  so  as  to 
di^^de  it  into  two  equal  parts. 

10.  To  inscribe,  in  a  given  triangle,  a  rectangle  whose  area  is 
known. 

11.  Through  two  given  points,  to  describe  a  circle  tangent  to 
a  given  right  line,     p 


e 


PART  II 


INDETERMINATE  GEOMETRY 


14.  The  second  branch  of  Analytical  Geometry,  wliicli  has  for 
its  principal  object  the  analytical  investigation  of  the  generjil 
properties  of  lines  and  surfaces,  is  much  more  extended  in  its  ap- 
plication, and  interesting  in  its  results,  than  that  which  we  have 
just  examined.  It  is  called  Indeterminate  Geometry,  from  the 
fact,  that,  in  the  equations  used,  the  unknown  quantities  admit  of 
an  infinite  number  of  values,  or  are  indeterminate,  and  are  there- 
fore called  variables ;  while  from  the  nature  of  the  problems  dis- 
cussed in  the  first  branch,  they  admit  of  a  finite  number  of  values 
only,  and  must  be  determinate. 

OF    POINTS    IN    A    GIVEN    PLANE. 


15.  Let  AX  and  AY  be  two  fixed  right  lines,  indefinite  in  ex- 
tent, and  M  any  point  of  their  plane  within  the  angle  YAX. 
Through  this  point  draw  MR  and  MP  parallel  respectively  to  AX 

and  AY ;  then  if  the  distances 
MR  and  MP  are  given,  it  is  evi- 
dent that  the  position  of  the  point 
M,  will  be  known,  and  may  be 
constructed,  by  laying  off  on  the 
line  AX,  beginning  at  A,  AP 
=  RiSI,  drawing  PM  parallel  to 
AY ;    then    on    AY,   laying   off 


INDETERMINATE    GEOMETRY.  25 

AR  =  PM  and  drawing  RM  parallel  to  AX  ;  the  point  of  in- 
tersection of  these  parallels  will  be  the  required  point. 

The  distances  MR  and  MP  are  called  the  rectilineal  co-ordinates 
of  the  point.  The  first,  or  the  distance  of  the  point  from  AY,  is 
the  abscissa ;  and  the  second,  or  the  distance  of  the  point  from 
AX,  is  the  ordinate  of  the  point,  these  distances  being  measured 
on  lines  parallel  respectively,  to  AX  and  AY. 

The  fixed  lines,  to  which  the  point  is  thus  referred,  are  ciilled 
the  axes  of  co-ordinates,  or  co-ordinate  axes. 

Their  point  of  intersection  A,  from  which  both  abscissa  and  or- 
dinate are  estimated,  is  the  or  if/in  of  co-ordinates. 


16.  The  abscissas  of  points,  the  position  of  which  is  indetermi- 
nate, are,  in  general,  denoted  by  the  letter  x,  and  the  ordinates  by 
y,  though  other  letters  are  sometimes  used. 

The  co-ordinates  of  points,  the  position  of  which  is  known,  are 
usually  denoted  either  by  the  first  letters  of  the  alphabet,  or  by 
the  symbols  x',  y',  x",  y",  &c.  If  we  denote  the  co-ordinates  MR 
by  a,  and  MP  by  h,  the  equations 

X  =  a  y  =   b..... (1), 

are  called  the  equations  of  the  point  M,  and  the  values  of  a  and  h 
being  known,  the  point  is  said  to  be  given,  and  may  be  con- 
btructed,  in  the  first  anyle,  YAX,  by  laying  off  AP  =  a  and 
AR  =  6,     as  in  the  preceding  article. 

If,  at  the  same  time,  w^e  consider  the  point  M',  having 
AP'  =  AP,  and  P'M'  =  PM,  it  becomes  necessary  to  adopt 
some  notation,  by  which  the  two  points  may  be  distinguished 
from  each  other.  This  notation  is  at  once  suggested,  by  a  re- 
ference to  that  which  is  used  in  a  similar  case,  for  the  cosine  of  an 
arc  in  Trigonometry,  and  the  abscissa  AP'  is  regarded  as  nef/ative 
Thus  the  equations  of  a  point  in  the  second  angle,  YAX',  are 

X  =    —   a  y  =   h. 


26  INDETERMINATE    GEOMETRY. 

If  tlie  point  is  below  the  axis  of  abscissas,  its  ordinate,  from 
analogy  to  the  sine  of  an  arc,  is  regarded  as  negative.  Thus  the 
equations  of  the  point  M",  in  the  third  angle  Y'AX',  are 

X  =    —   c^  2/=    —   ^  \ 

and  in  like  manner,  the  equations  of  the  point  M'",  in   tlie  fourth 
angle ^  Y'AX,  arc 

•T  =  a  y  =   —  ^' 

Thus  it  appears,  that  by  assigning  proper  values  and  signs  to  a 
and  5,  equations  (1)  may  be  regarded  as  the  representatives  of  any 
point  in  the  plane  of  the  co-ordinate  axes. 

If  the  point  is  on  the  axis  of  X,  (the  axis  of  abscissas),  its  ordi- 
nate must  be  0,  and  its  equations 

X  ■=  a  y  z=   0. 

If  it  is  on  the  axis  of  Y,  its  abscissa  must  be  0,  and  its  equations 
a;  =   0  y  =z   h. 

By  the  essential  signs  of  a  and  ft,  in  these  equations,  we  ascer- 
tain whether  the  points  are  on  the  right  or  left  of  the  origin,  above 
or  below  the  axis  of  X. 

[f  the  point  is  on  both  axes  at  the  same  time,  that  is,  at  the 
origin,  its  equations,  or  the  equations  of  the  origin^  become 

X   =   0  y  =   0. 


'.     Let  x',  y\  and  x"y  y"^  be  the  co-ordinates  of  any  two 

Y  points,   as  M  and  M',  in  the  plane 

/  ^^^     YAX.     Join  M  and  M',   and  draw 

/   ^,^£i:rrlir../'i2  MR  parallel  to  AX,  then  in  the   tri- 

/       /  /  angle   MM'R   we  have,  from  Triq^o- 

—jf jp—X  nometry, 

MM'  =  ^/mR**  -f-  M^'  -  2MR  x  M'R  cos  MRM' 


INDETERMINATE    GEOMETRY. 

the  radius  being  supposed  equal  to  unity.     But 


27 


MR 


PP'   ■=  x"  —  x' 


M'R  =  v"  — 


/; 


hence,  denoting  MM 'by  D,  the  angle  "5CAX  by  ^,  and  observing 
that    cos  MRM'  =   —  cos  /3,    we  have 

D  =  V(x"  —  x'Y  +  {y"  —  y'f  +  2(a;"  -  x') {y"  -  y )  cos  /3...(l). 
If    /3  =   90°,     cos  /3  =   0,     and  this  formula  reduces  to 

(2), 


D  =  V{x"  -  ^0'  +  {y"  -  y'] 


that  is,  if  the  axes  of  co-ordinates  are  perpendicular  to  each 
other,  the  distance  between  two  points,  in  their  plane,  is  equal  to  the 
square  root  of  the  sum  of  the  squares  of  the  differences  of  the  ab- 
scissas and  ordinates  of  the  points. 

If  one  of  the  points,  as  M,  is  at  the  origin,  x'  and  y'  will  be  0, 
and  the  last  formula  reduce  to 


D  =   fx"^  + 


If^  Let  P  be  a  fixed  point,  PS  a  fixed  right  lifte,  and  M  any 
point  of  a  plane  containing  PS.  If  the  length  of  the  line  PM,  which 
we   represent  by   ?',  and  the  , 

angle  v,  made  by  this  line 
^vith  the  fixed  line,  are  given, 
the  position  of  the  point  will 
be  fully  determined,  and  may 
be  constructed,  by  drawing 
through  P  a  line  raaliing,  with  the  line  PS,  the  given  angle,  and 
then  from  the  point  P,  laying  off,  on  this  line,  the  given  distance. 
By  varying  the  angle  v,  through  all  values  from  0  to  360°,  and 
the  line  r  from  0  to  infinity,  the  position  of  every  point  of  the 
plane  may  be  determined. 

Tlie  point  P  is  called  the  pole  ;  the  line  PM,  the  radius  vector^ 
and  the  variables  r  and  v,  the  polar  co-ordiiiates  of  the  point. 


28 


INDETERMINATE    GEOMETRY. 


19.  By  a  renew  of  tlie  preceding  discussion,  we  see  that  the 
position  of  the  points  have  been  determined  by  ascertaining  their 
situation  with  reference  to  certain  other  fixed  points  or  magnitudes. 
In  the  first,  or  system  of  rectilineal  co-ordinates^  the  points  are  referred 
to  two  fixed  right  lines,  and  the  means  of  reference  are  two  other 
right  lines,  which  vary  in  length,  as  the  position  of  the  point  is 
changed.  In  the  second,  or  system  of  polar  co-ordinates,  the  points 
are  referred  to  a  f.xed  point  and  a  fixed  right  line,  and  the  means 
of  reference  are  a  variable  angle  and  a  variable  right  line. 

Although  there  are  other  methods  of  determining  the  position 
of  points,  these  are  the  two  in  most  general  use.  In  every  system, 
it  'should  be  observed,  that  the  position,  thus  determined,  is  not 
absolute  but  relative,  as  all  that  thus  becomes  knoAvn,  is  the 
position  of  the  point  with  reference  to  some  other  points  or  mag- 
nitudes ;  and  also,  that  the  general  name  of  co-ordinates  of  a  jwint, 
is  applied  to  tlie  elements,  of  whatever  nature,  by  means  of  which 
the  position  of  the  point  is  determined. 


OF    THE    RIGHT    LINE    IN    A    GIVEN    PLANE. 

20.     Let  BM  be  any  right  line,  in  the  plane  of  the  co-ordinate 
axes  AX  and  AY,  and  let  M  be  any  point  of  the  line,  of  which 

the  co-ordinates  AP  and  MP 
/Y  ^T  -^  a^'6  denoted  by  x  and  y. 
Throuifh  the  orio-in  A,  draw 
AM'  parallel  to  BM.  Re- 
present the  angle  YAX  by  ^, 
and  MBX  =  M'AX  by  a ; 
the  angle  PM'A  =  M'AY  will 
then  be  represented  by  (3  —  a. 
BVom  the  triangle  AM'P,  we  have  the  proportion 

AP  :  PM'  :  :  sin  PM'A :  sin  M'AP    or  : :  sin  (^  —  a)  :  sin  a 


or,  representing  PM'  by  y' 


1 

INDETERMINATE    GEOMETRY.  29 

X  '.  y'  :  :  sin  (/3  —  a)  :  sin  a, 

and  since  AP  is  to  PM'  as  the  abscissa  of  any  other  point  of  the 
line  AM'  is  to  its  corresponding  ordinate,  this  relation  will  exist 
whatever  be  the  position  of  the  point  M',  on  the  line  AM'.  From 
this  proportion,  we  deduce 

,  sin  a 

y'  = 


sin  (/3  —  a) 

But  ihe  ordinate  of  any  point  of  the  line  BM,  as  PM,  exceeds  the 
corresponding  ordinate  of  the  line  A^',  by  the  constant  distance 
MM''  =  AC.  Representing  this  distance  by  6,  we  have,  for  every 
point  of  tlie  line, 

/    I    /,                                          sin  a  ,    T 

y  =^.y'  +  0,  or  y   =z    -—^ -X   +  6. 

sm  (p  —  a) 

This  equation  expresses  the  relation  between  the  co-ordinates, 
X  and  y,  of  every  point  of  the  line  BM,  and  is  called  ihe  equa- 
tion of  the  line.  The  co-efficient  of  ar,  in  this  equation,  represents 
the  ratio  of  the  sines  of  the  angles  which  the  line  makes  with  the 
axes  of  X  and  Y,  and  the  absolute  term,  (6),  the  distance  from  tlie 
origin  to  the  point  in  which  the  line  cuts  the  axis  of  Y. 


21.     By  attributing,  in  succession,  all  values  to  a,   between  0 

and  360°,  and  all  possible  values  to  6,  both  positive  and  negative, 

the  equation 

sin  a              ,     ,  f^^ 

y  =  -T-.^ r^  +  ^ (IX 

sm  (/3  —  a) 

may  be  made  to  represent  every  right  line  in  the  plane  of  the  co 
ordinate  axes. 

If  h  is  negative^  the  line  takes  the  position  B'C,  and  its  equation 

will  be 

y  T=   X  —  6. 

^  .in   (/5  -  a) 


i' 


i'.'d  IND  /I'ERMIXATE    GEOMETRY. 

If  a  =  0,  the  line  is  parallel  to  the  axis  of  X,  and  its  equation 
reduces  to 

y   =    0.x   '\-    b, 
or 

y  =   6,  X  being  indeterminate. 

If  6  =  0,  at  the  same  time,  the  line  coincides  with  the  axis  of  X ; 
lience,  the  equation  of  the  axis  of  X,  is 

2/  =   0,  X  being  indeterminate^ 

If  b  is  positive,  the  line  is  above  the  axis  of  X ;  if  negative,  it  is 
below  it. 

Solving  equation  (1)  ^vith  respect  to  x,  it  is  put  under  the  form 


_   sin  (/3  —  a)  5  sin  (/3  —  a)  ,^^ 

sm  a  sm  a 

From  the  triangle  BAG,  we  have 

b  sin  (/3  —   a) 
sin  a  :  sin  (,/3  -  a)  :  :  b  :  AB  =    ^^ > 

that  is,  the  absolute  term  of  equation  (2),  represents  the  distance 
from  the  origin  to  the  point  in  which  the  line  cuts  the  axis  of  X. 
Representing  this  by  a,  the  equation  becomes 

sin  (/3  —   a)  „.    ,     „ 
sin  a 

If  in  this     a  =  /3,    the  line  is  parallel   to  the  axis  of  Y,  and 
the  equation  reduces  to 

a;  =   O.y  +  cr, 

or 

X  =  a,  y  being  indeterminate. 

If    0  =  0,     at  the  same  time,  the  line  coincides  with  the  axis 
of  Y,  and  its  equation  becomes 


INDETERMINATE    GEOMKTR*  31 

X  =  0,  y  helng  indeterminate. 

If  a  is  positive,  the  line  is  on  tlie  right,  if  negative,  it  is  on  the 
left  of  the  axis  of  Y. 


'22.  Equation  (1),  of  the  preceding  article,  contains  two  kinds 
of  quantities,  x  and  y,  which  are  different  for  different  points  of  the 
line,  and  are  therefore  called  variables  ;  and  a,  /3  and  5,  which  re- 
main the  same  for  the  same  line,  and  are  called  constants.  When 
the  values  of  these  constants  are  knawn,  the  line,  as  Ave  have  seen, 
is  fixed  in  position,  or  completely  determined. 

Since  the  equation  contains  two  variables,  we  may  assign  any 
value  to  one,  and,  by  the  solution  of  the  equation,  deduce  the  cor- 
responding value  of  the  other.  These  two,  taken  together,  will  be 
the  co-ordinates  of  a  point  of  the  line,  which  may  be  constructed 
as  in  Art.  (15).  By  assigning  other  values,  in  succession,  to  one 
of  the .  variables,  and  deducing  the  corresponding  values  of  the 
second,  any  number  of  points  may  be  determined,  and  the  line  h( 
thus  constructed  hy  points. 

Likewise,  if  either  of  the  co-ordinates  of  a  point  of  the  lino  is 
known,  the  other  may,  at  once,  be  deduced,  by  substituting  the 
known  value  in  the  equation,  and  solving  it  with  reference  to  the 
variable  w^hose  value  is  required.  Thus,  we  know  that  the  ab- 
scissa of  that  point  of  the  line,  which  is  on  the  axis  of  Y,  is  0.  Sub- 
stituting a;  =  0,  in  the  equation,  we  deduce  y  =  h^  which 
is  the  ordinate  of  liic  point  in  which  the  line  cuts  the  axis  of  Y. 

If  we  substitute  y  =  0,  the  resulting  value  of  x,  will  be  the 
abscissa  of  the  point  in  which  the  line  cuts  the  axis  of  X.  This 
ordinate  and  abscissa  being  laid  off  respectively  on  the  axes  of  Y 
and  X,  a  right  line,  drawn  through  their  extremities,  will  be  the 
line  to  which  the  equation  belongs. 

23.     F.-om  the  preceding  article   we  see  that  equation   (1)  of 


32  INDETERMINATE    GEOMETRY. 

Art.  (21)  is  truly  the  analytical  representative  of  the  right  Hne. 
And  in  general,  any  line,  curved  as  well  as  straight,  may  be  thus 
represented  by  its  equation  ;  that  is,  hy  an  equation  which  ex- 
presses the  relation  between  the  co-ordinates  of  every  point  of  the 
line. 

Every  such  equation  -will  contain  two  kinds  of  quantities,  viz  : 
variables  and  constants.  The  variables  represent  the  co-ordinates 
of  the  different  points  of  the  line,  and  the  constants  serve  to  deter- 
mine its  position  and  extent.  This  is  plain  from  the  fact,  that  the 
constants  being  given,  all  the  points  of  the  line  may  be  construct- 
ed, as  in  Art.  (22).  We  therefore  say,  that  a  line  is  given,  when 
the  form  of  its  equation^  and  the  constants^  which  enter  it,  are 
known. 

From  the  definition  of  the  equation,  it  follows,  that  if  a  iwint  is 
on  a  given  line,  its  co-ordinates,  when  substituted  for  the  variables, 
must  satisfy  the  equation  of  the  line. 

Also,  if  a  point  is  not  on  a  given  line,  its  co-ordinates  loill  7ioi 

"■*""""'■"■  ■  ^ 

24.  It  will,  in  general,  be  found  more  convenient  to  take  the 
co-ordinate  axes  at  right  angles  to  each  other ;  and  they  will  be  so 
regarded,  unless  it  is  otherwise  expressly  mentioned.  Under  this 
supposition,  /3,  in  equation  (1)  of  Art.  (21),  will  be  90°, 

sin  (/3   —   0?)   =  sin  (90°   —   a)   =  cos  a, 

and  the  equation  reduce  to 

y  =  !!!!-?  X  ^  b,         or         y  =  tang  ^  -f  6,  * 
cos  a 

or  denoting  tang  a  by  a 


*  Note.— In  Analytical  Geometry,  R  or  the  radius  of  the  trigonometri- 
cal tables,  is  always  regarded  as  linily,  unless  it  is  otherwise  mentioned. 


INDETERMINATE    GEOMETRY. 

ax   +   ^^ 


y 


(1), 


in  wliioli,  it  should  be  remembered,  that  a  represents  the  tangent 
of  the  angle  which  the  line  makes  with  the  axis  o^  abscissas,  and 
h  the  distance  from  the  origin  to  the  point,  in  which  the  line  cuts 
the  axis  of  ordinates. 

If  the  line  passes  through  the  origin,     5  =   0,    and  the  equa- 
tion becomes 

y  =  ax. 

If  the  line  occupies  the  position  of  AM,  the  angle  a  being  acute, 
a  ]s  positive,  and  as  for  all  points  of  the  line  in  the  first  angle,  x  is 
also  positive,  the  product,  ax,  is  posi- 
tive, as  it  should  be,  since  y  must  be 
positive  for  all  points  above  AX.  For 
all  points  in  the  third  angle,  x  being 
negative,  ax  is  negative,  as  it  should 
be,  since  y  must  be  negative  for  all 
points  below  AX. 

If  the  line  occupies  the  position 
AM' ;  as  the  angle  a  is  estimated  from  the  axis  of  X,  on  the  right 
of  the  line,  around  to  it,  as  indicated  in  the  figure ;  a  is  obtuse 
and  a  negative.  For  all  points  of  the  line  in  the  second  angle,  x 
is  negative  and  ax  positive.  For  points  in  the  fourth,  x  is  positive 
and  ax  negative. 


25.     Every  equation  of  the  first  degree,  between  two  variables, 
will  be  a  particular  case  of  the  general  form 

Aar  4-  By  +  C  =  0, 

and  this,  when  solved  with  reference  to  y,  gives 


^  B  "^        B 


an  equation  of  the  same  nature  and  form  as 
3 


34    ,  INOETEUMINATE    GEOMETRY. 

y   =   ax    -{•    h, 

and  may  therefore  be  regarded  as  the  equation  of  a  right  line,  in 

which,  (the  axes  of  co-ordinates  being  at  right  angles,)      —  — 

13 

will  represent  the  tangent  of  the  angle  which  the  line  makes  with 

the  axis  of  X,  and    —  —     the  distance  from  the  origin  to  the 
B 

point  in  which  the  line  cuts  the  axis  of  Y. 

If  the  equation  be  solved  with  reference  to  x^  it  will  appear 

under  the  form 

X  =  a'y  H-  h' 

fin  which  a'   =  _   will  be  the  tangent  of  the  angle  made  with  the 
a 

axis  of  Y,  and  V  the  distance  cut  off  by  the  line  on  the  axis  of  X. 
Hence,  every  equation  of  the  first  degree,  between  two  variables,  re- 
presents a  right  line  ;  and  if  it  be  solved  with  reference  to  either 
variable,  the  coefficient  of  the  other  will  be  the  tangent  of  the  angle, 
which  the  line  makes  with  the  axis  of  that  variable  ;  and  the  ab- 
solute term  will  be  the  distance  cut  off,  by  the  line,  on  the  axis  of 
that  vai'iable,  with  reference  to  which  the  equation  is  solved. 


26.  The  manner  of  constructing  a  right  line,  from  its  equation, 
may  be  illustrated  by  the  following 

E.TampIes. 

I.  Take  the  equation 

21/  -   4x  +   3   =  0. 

Making     rr  =  0,     wo  deduce     y  =    —  |,     for  the  ordinate 
of  the  point,  in  which  the   line  cuts  the  axis  of  Y,  Art.  (22). 


INDETERMINATE    GEOMETKY. 


35 


Making    y  =  0,     we  deduce     a?  =  |,     for  the  abscissa  of  the 
point,  in  which  the  line  cuts  the  axis  of 
X.     Assuming  any   convenient  unit  of 
length,  and  laying  off    AC  =  —  f ,  and 

AB  =   f,  BC  will  be   the   hne   repre-      

sen  ted  by  the  equation. 

Or,  thus  :   Solving  the  equation  with 
reference  to  y,  we  have 


y  =  2.1; 


Laying  off  the  distance  AC  =  —  |,  and  drawing  the  hne 
CB,  making,  with  the  axis  of  X,  an  angle  whose  tangent  is  2,*  it 
will  be  the  hne. 

Or  the  line  may  be  constructed  by  points,  thus  :  Making 


a:  =   1 


we  deduce 


y  = 


X  =  2 


&c. 


The  points,  represented  by  the  different  sets  of  co-ordinates  thus 
determined,  may  be  constructed  and  the  line  drawn  through  them. 

2.  3y  +  9.r  -   1   =  0. 


*  Note. — An  angle  whose  tangent  is  a  given  number  ^ 

may  always  be  constructed  thus.     Let  tang  az=  -.    Lay 

d 

ujf   AB  =  d  and  erect  the  perpendicular  BM  =  c ;  draw 

AM,  the  angle  MAB  will  be  the  required  angle.    For  we 

have  A    d_ 

tang  MAB  =  IM  =  £  =  tang  a. 
^  AB        d! 

When  tang  a  is  a  whole  number,  as  in  the  example,    d  =  AB  =  1 

Vif.  unit  of  Icns^th. 


36  INDETERMINATE    GEOMETRY, 

3.  7/    —    X    —    4    =    0. 

4.  2i/  +   3x   +   5   =   0. 

27.     Let  y  =  ax  +  h 

y  =   a'x   +   h' 

be  the  equations  of  two  riglit  lines.  Those  values  of  x  and  y 
which,  taken  together,  will  satisfy  both  of  these  equations,  must  be 
the  co-ordinates  of  a  point  on  each  line,  Art.  (23).  But  if  we  com- 
bine the  two  equations  and  deduce  the  values  of  x  and  y,  we  ob- 
tain all  which  can  possibly  satisfy  both  equations  at  the  same 
time  ;  these  values  must  then  be  the  co-ordinates  of  all  points 
common  to  the  lines. 

Placing  the  second  members  of  the  equations  equal,  we  have 

ax  -{■   b  =  a'x  -f  6', 

whence 

y  -  b 

X    =z    . 


Substituting  this  value  of  x,  in  the  first  equation,  we  obtain 

/               ab'  -  a'b 
J     y   = . 

a  —  a' 

These  values  of  x  and  y  must  be  the  co-ordinates  of  a  point 
common  to  both  lines.  And,  in  general,  since  the  equations  of 
right  hnes  are  of  the  first  degree,  the  values  resulting  from  their 
combination  must  be  real  and  give  one  common  point,  and  only  one. 

If  a  =  a',  the  values  of  x  and  y,  both  reduce  to  infinity ; 
the  point  of  intersection  is  then  at  an  infinite  distance,  that  is,  the 
lines  are  parallel. 

U    b  =  b\     at  the  same  time,  both  values    become  -,  or  in- 

0 
determinate,  as   they   should,   since   in  this   case  the  two  lir^fl 

coincide  and  have  all  their  points  common. 


INDETERMINATE    GEOMETRY. 


37 


It  IS  evident  that  the  above  reasoning  will  apply  to  any  lines, 
straight  or  curved,  and  we  may  therefore  give  the  following  rule 
for  obtaining  the  points  of  intersection  of  any  two  lines.  Combine 
the  equations  of  the  lines,  and  deduce  the  values  of  the  variables. 
For  each  couple  of  real  values  there  will  be  a  common  point.  If 
the  values  are  all  imaginary,  there  will  be  no  common  point. 

Find  the  point  of  intersection  of  the  two  right  lines  given  by 
the  equations 

3w  -  2a;  +   1 


28.  Let 


be  the  equations  of  any  two  given  right 
hues,  making  the  angles  a  and  a',  respec- 
tively, with  the  axis  of  X,  and  the  angle 
V  with  each  other.  By  the  figure,  we 
see  that 

a'   =  V  +    a,     or      V   =  a'   —   a. 


=  0,                  5y  +   dx  =  0. 

y  ^z  ax   +   b 
y  =z  a'x  -{-   b' 

r 

y/ 


and  by  the  trigonometrical  formula,  for  the  tangent  of  the  differ- 
ence of  two  angles, 

tanc.  V  =     tang  a'  -  tang  a 


1   +   tang  a'  tang  a 
and  since  from  the  equations  of  the  lines.  Art.  (24), 
a  =  tang  a  a'  =  tang  a', 


have 


tang  V  = 


a'  —  a 


I   +  a'a  ' 
from  which,  by  the  substitution  of  the  values  of  a'  and  a,  given 


38  INDETERMINATE    GEOMETRY. 

by  particular  equations,  the  natural  tangent  of  the  angle  between 
two  right  lines  may  be  found  ;  and  from  a  table  of  natural  sines, 
cosines,  <fec.,  the  value  of  the  angle,  in  degrees,  minutes,  &c.,  may 
be  determined. 
If    a  =   a', 

tang  V  =  °   ,      =  0 ; 

1    +   a'a 

hence,  in  this  case,  the  angle  is  0,  and  the  lines  are  parallel,  as 
shown  in  the  preceding  article. 
If     1   +   aa'  =  0, 

.         ,7.         a'  —  a 

tang  V   =:  =  00, 


the  angle  is  90°,  and  the  two  lines  are  perpendicular  to  each 
other. 

To  ascertain  then,  practically,  whether  two  right  lines  are  par- 
allel or  perpendicular  :  Solve  their  equations  with  reference  to 
either  variable  ;  if  the  coefficients  of  the  other  variable  are  equal, 
the  lines  are  parallel ;  if  the  product  of  these  coefficients  plus 
nniiy  is  equal  to  0,  they  are  perpendicular. 

Apply  this  rule  to  the  equations 

1.  2y  —  4a:  +   Y   =   0,  y  —   2a;  —  3   =   0. 

2.  2/  —   3a;  +    1   =  0,         Gy  +   2a;  —  5  =   0. 

29.     Let  x'  and  y'  be  the  co-ordinates  of  a  given  point,  and 

y  ^  ax  +   b (1), 

the  general  equation  of  a  right  ^ine,  in  which  a  and  b  are  undeter- 
mined. If  the  given  point  is  on  the  line,  its  co-ordinates,  when 
substituted  for  x  and  y,  must  satisfy  the  equation.  Art.  (23),  and 
wo  must  have 

y'  =  ax'  +  h, 


INDETERMINATE    GEOMETRY.  3& 

an  equation  expressing  the  condition  that  the  point,  (.t',  y'),  shall 
be  on  the  line.  This  condition  may  be  introduced  into  equation 
(1)  by  subtracting  it,  member  from  member.     We  thas  obtain 

y  —  y'   =  a{x  —  x') (2), 

which  is  the  equation  of  a  right  line  with  the  condition  introduced, 
that  a  given  point  shall  be  on  it;  or,  is  the  equation  of  a  right 
line  passing  through  a  given  point. 

a  remains  undetermined,  as  it  should,  since  an  infinite  number 
of  right  lines  may  be  drawn  through  the  given  point.  If  the 
abscissa  and  ordinate  of  the  given  point  are  2  and  3,  the  equation 
of  the  line  becomes 

y  —    3   =   a(a;  —    2). 


30.  If  the  line,  represented  by  equation  (2)  of  the  preceding 
article,  be  subjected  to  the  condition  that  it  shall  be  parallel  to  a 
given  line,  as  the  one  whose  equation  is, 

y  =  a'x  +  h, 

we  must  have.  Art.  (28), 

a  =  a'. 

Substituting  this  known  value  in  equation  (2),  the  line  will  be 
fixed  and  its  equation  become 

y  —  y'   =  a'{x  —  x'). 

K  the  line  is  required  to  be  perpendicular  to  the  given  lino,  wa 
nust  h'ive 

1   -f  aa'  =  0,  or  a  =  —  — , 

ttf 


and  the  equation  becomes 


40 


INDETERMINATE    GEOMETRY. 


y'  =   -  i  (^  -  X'). 


1.  Find  the  equation  of  a  right  I'ne,  passing  through  the  point, 
r/  —   —   2,3/'  =  3,  and  parallel  to  the  line  whose  equation  is 

2y  —  X  +  2  =  0. 

2.  Find  the  equation  of  a  right  line,  passing  through  the  same 
point  and  perpendicular  to  the  same  line. 

31.  If  the  line  represented  by  equation  (2),  Art.  (29),  be  sub- 
jected to  the  condition,  that  it  shall  pass  through  another  point 
whose  co-ordinates  are  x"  and  y",  these  co-ordinates  must  satisfy 
the  equation  and  give  the  equation  of  condition  u^ii*-  u^ii  ^i-*) 

y"  —  y'  =  a{x"  —  x'), 


from  which,  the  value  of  a  becomes  known,  and  we  have 

.      y"  -  y' 


Substituting  this,  in  equation  (2),  we  obtain 


•(1), 


for  the  equation  of  a  right  line  passing  through  two  given  points. 
Y  . ,  If  M  and  M'  are  the  points,  the  co« 

ordinates  of  the  first  being  x'^  y\  and 
\j^         of  the  second  x",  y'\  we  have 

M'R  =  y'  -  2/',     MR  =  a;"  —  x\ 


T    Ji     B 


JP'  X 


tang  M'MR  =  tang  M'TX  =  M!5  =  y"  ""  ^^    =  g. 
^  °  MR         x"  -  ar' 


INDETERMINATE    GEOMETRY.  41 

If  y''  =  y\  this  value  of  a  reduces  to 

0 


a  = 


X"     —    X' 


as  it  should,  since  the  line  becomes  parallel  to  the  axis  of  X. 
If    x"  =  a/, 


y"  —  y' 


as  in  this  case  the  line  is  peri)endicular  to  the  axis  of  X. 
If    x"  =  x'     and    y"  =  y', 

a  =  -     indetermiTmte. 
0 

since  the  two  points  become  one,  through  which  an  infinite  num- 
ber of  right  lines  may  be  drawn. 

1.  If  the  co-ordinates  of  the  points  are    a;'  =   2,    2^'  =   — -   1 ; 
x''  =   3,     y"  =  0  ;     equation  (1)  will  become 


2/   +    1   =  i  (.r  -   2),  "f^ 


1 

which  reduces  to 

y  =  a;  —   3. 

2.  Find  the  equation  of  a  right  line  passing  through  the  two 
ix)ints 

«'  =   —   !,         y'  =    -  2;  x"  =  A,        y"  =   -  5. 


32.  In  every  equation  containing  but  two  variables,  we  may, 
as  in  Art.  (22),  assign  to  one  a  series  of  values,  in  succession, 
and  deduce  the  corresponding  values  of  the  other,  and  thus  con- 
struct a  series  of  points,  which  being  joined,  will  evidently  form  a 
line,  which  will  be  represented  by  the  given  equation.     Hence  we 


42  INDETERMINATE    GEOMETRY. 

say,  in  general,  ^hat  every  equation  between  two  variables^  is  tJis 
equation  of  a  line^  either  straight  or  curved. 

If  all  values  of  the  first  variable  give  imaginary  values  for  the 
second,  the  line  is  said  to  be  imaginary. 

If  there  is  but  a  limited  number  of  couples  of  real  values,  which 
will  satisfy  the  equation,  it  will  represent  a  point  or  a  limited 
number  of  distinct  points. 


33.  Whenever  the  relation  between  the  ordinate  and  abscissa 
of  a  line  can  be  expressed  by  the  ordinary  operations  of  Al- 
gebra, that  is,  by  addition,  subtraction,  multiplication,  division, 
the  formation  of  powers  denoted  by  constant  exponents,  or  the  ex- 
traction of  roots  indicated  by  constant  indices,  the  line  is  said  to 
be  Algebraic, 

When  this  relation  can  not  be  so  expressed,  the  line  is  Trans- 
cendental. 

Algebraic  lines  only,  will  be  considered  in  this  Treatise.  They 
are  classed  into  orders,  according  to  the  degree  vf  their  equations. 
Thus,  a  line  of  the  first  order,  is  one  whose  equation  is  of  the  first 
degree.  A  line  of  the  second  order,  one  whose  equation  is  of  the 
<iecond  degree,  <fec.  We  have  seen.  Art.  (25),  that  the  right  line  is 
the  only  line  of  the  first  order. 

The  discussion  of  the  equation  of  a  line  consists  in  classing  the 
line,  determining  its  form,  its  limits,  its  position  with  respect  to 
the  co-ordinate  axes,  and  the  points  in  which  it  cuts  these  axes. 


OP   THE    CIRCLE. 

34.  Let  x'  and  y'  be  the  co-ordinates  of  the  centre  of  a  circle, 
and  R  its  radius,  and  let  x  and  y  be  the  co-ordinales  of  any  point 
of  its  circumference.  The  distance  from  the  centre  to  any  point 
of  the  circumference,  will  then,  Art.  {11),  be  denoted  by 


INDETERMINATE    GEOMETRY. 


43 


V{x  -  x'Y  +  (y  -  9/y ; 


but,  from  the  definition  of  a  circumference,  this  distance  must  be 
constantly  equal  to  the  radius,  R  ;  hence  we  have 


Of 


V(x  -  x'Y  +  (y  -  y'Y  =  R, 


(x  -  x'Y  +  (y  -  yy  =  K« (1) ; 


and  since  this  expresses  the  relation  between  the  co-ordinates  of 
every  point  of  the  circumference,  Art.  (23),  it  is  the  equation  of 
the  circumference,  or  the  equation  of  the  circle ;  the  word  circle 
being  commonly  used  for  cu'cumference. 

The  circle  will  be  given,  when  x\  y',  and  II  are  given,  Art.  (23), 
and  by  attributing  different  values  to  these  constants,  we  may 
place  the  centre  in  any  position,  and  give  to  the  circle  any  extent. 

For  those  points  of  the  circle  which  lie  oh  the  axis  of  X,  y  =  0 ; 
substituting  this  in  equation  (1),  the  corresponding  values, 

X  =   x'   ±:    -v/Ra   _   y>2^ 

will  be  the  abscissas  of  the  points,  in  which  the  circle  cuts  the  axis 
ofX. 

If  y'  <  R,  these  values  will  be  real,  and  the  circle  will  in- 
tersect the  axis,  in  two  points. 

If  y'  =  R,  the  two  points  will  unite,  and  the  circle  will  be 
tangent  to  the  axis  of  X. 

If  ?/'  >  R,  the  values  of  x  will  be  imaginary,  and  there  will 
be  no  point  of  intersection. 

Each  position   of    the   circle    is 
shown,  in  the  accompanying  figure. 

By  making    x  =  0,    we  deduce 

y  -   7/  ±:    VR^  -   x'^, 

for  the  ordinates  of  the  points,  in 
which  the  circle  intersects  the  axis 


44  INDETERMINATE    GEOMETRY. 

of  y,  and  these  will  be  real,  equal,  or  imaginary,  according  as  x 
less,  equal  to,  or  greater  than  R. 

Solving  equation  (1)  with  reference  to  i/,  we  have 


y  =  y'  ±:   VW^^^Jx~^^^^~7^, 

By  assigning  values  to  a:,  in  succession,  we  deduce  the  corres- 
ponding values  of  y,  and  thus  determine  as  many  points  of  the 
curve  as  we  please,  Art.  (32). 

Every  value  of  x,  which  makes  {x  —  x'Y  <  R^  will  give 
two  real  values  of  y.  For  every  such  value,  there  will,  conse- 
quently, be  two  corresponding  points  of  the  curve. 

If  a:  =  a;'  +  R  or  x'  —  R,  the  values  of  y  will  be 
y'  ±  0  ;  the  two  points  will  unite,  and  the  corresponding  ordi- 
nate will  be  tangent  to  the  curve,  as  SM  or  S'M'. 

If  ar  >  a;'  +  R  or  <  a;'  —  R,  the  values  of  y  will  be 
imaginary,  and  there  will  be  no  corresponding  points  of  the 
curve. 

We  thus  see  that  the  curve  is  limited,  in  the  direction  of  the 
axis  of  X,  by  the  two  lines,  SM  and  S'M'.  In  the  same  way,  by 
solving  the  equation  with  reference  to  x,  we  may  obtain  the  limits 
in  the  direction  of  the  axis  of  Y. 


35.  If  x'  and  y'  are  both  equal  to  0,  the  centre  of  the  circle 
will  be  at  the  origin  of  co-ordinates,  and  equation  (1),  of  the  pre- 
ceding article,  will  reduce  to 

ar2  -f  2/3  ^   R« (1). 

To  discuss  this  equation.  Art.  (33) ;  make  y  =  0,  we  thus 
obtain 

a;  =   ±  R, 

which  shows,  that  the  curve  cuts  the  axis  of  X,  in  the  two  points, 
B  and  C,  at  distances,  on  the  right  and  left  of  the  origin,,  each 


INDETERMINATE    GEOMETRY. 


equal  to  R. 
Making:    x 


=   0,    we  obtain 
=   ±  R, 


^ 

, 

I 

t 

1 

~~c 

^1 

E 

f 

i 

vvliicli  shows  that  the  curve   cuts 'the 
axis  of  Y,  in  the  two  points  D  and  E. 

Solving  the  equation  with  reference 
to  y,  we  have 

y   =    db    VR^   --~^, 


from  which,  we  see  that  every  value  of  x^  positive  or  negative,  and 
numerically  less  than  R,  gives  two  real  values  of  y,  equal  with 
contrary  signs  ;  hence,  for  each  of  these  values  there  are  two  cor- 
responding points,  one  above,  and  the  other  below  the  axis  of  X, 
at  equal  distances  from  it,  and  the  ordinates  of  these  points,  taken 
together,  form  a  chord,  which  is  bisected  by  the  axis  of  X.  This 
proves  that  the  curve  is  symmetrical  with  respect  to  the  axis  of  X. 

If  a;  =  rfc  R,  y  becomes  equal  to  ±  0,  which  proves  that 
the  corresponding  ordinates,  produced,  are  tangent  to  the  curve. 

If  X  is  numerically  greater  than  R,  either  positive  or  negative, 
the  values  of  y  are  imaginary,  and  there  are  no  corresponding 
points  of  the  curve.  The  curve  is  therefore  limited  in  the  direction 
A  the  axis  of  X,  by  the  two  tangents  at  B  and  C. 

In  a  similar  way,  it  niay  be  proved,  that  the  curve  is  symmetri- 
cal with  respect  to  the  axis  of  Y,  and  that  its  limits  are  two  tan- 
irents  at  D  and  E. 


J^ 


36.     For  every  point  of  the  curve,  as  M,  in  the  figure  of  the 
preceding  article,  we  have 

7  y2  =  R2  -  a2  =   (R  -f  x)  (R  -  a:)  =  CP  X  PB, 

a  well  known  property  of  the  circle. 


•X 


46  INDETERMINATE    GEOMETRY. 

37.  If  y  represents  the  ordinate  of  any  point,  as  M',  without 
the  circle,  in  the  figure  of  Art.  (35),  we  have 

^'    M'P  >  MP         or         2/3  >  R2  —  x\ 
For  any  point,  as  M",  within  the  circle,  we  have  '  '  ■■  ' 

M"P  <  MP        or        y2  <  R2  _  ^.a 
Hence  we  deduce  the  thre3  analytical  conditions  f 

y^  •\-  x^  —  R*  =  0,        for  a  point  on  the  circle^ 
y  y2  +  a;2  —  R2  >   0,  "  "     without  the  circle^ 

,    .  y8  -^.  a;2  —   R2  <   0,  "  "     loithin  the  circle. 

38.  If  the  origin  of  co-ordinates  is  at  0,  in  the  figure  of  Art 
(35),  the  co-ordinates  of  the  centre  wnll  be 

a;'  =  R  y'   =   0, 

and  the  general  equation  (1),  Art.  (34),  will  reduce  to 

a;2  _|-  y2  _   2Rx  =   0,         or         ?/«  =   2Rx  —  x\ 

This  equation  has  no  absolute  term,  or  term  independent  of  x 
and  y ;  the  substitution  of    a;  =  0    and    y  =  0,     will  there- 
in      fore  satisfy  it,  which  verifies  the  fact,  that  the  origin  of  co-ordinates 
/        is  on  the  curve ;  and,  in  general,  if  the  equation  of  a  line  has  no 
absolute  term^  the  line  passes  through  the  migin  of  co-ordiruites, 

OP    POINTS    IN    SPACE. 

39.  By  space  is  to  be  understood,  that  infinite  extent,  in  which 
all  bodies  are  situated.  As  the  absolute  places  of  points  and  mag- 
nitudes, in  this  indefinite  space,  can  not  be  determined,  we  have 
only  to  seek  their  situation,  with  reference  to  certain  other  objecte, 


» 


INDETERMINATE    GEOMKTRy.  47 

which  do  not  change  their  position  with  respect  to  each  other.  In 
a  plane,  we  have  seen  that  the  situation  of  points  and  hnes,  is  thus 
determined  by  a  reference  to  two  fixed  objects,  Art.  (19).  In 
space  it  is  found  necessary  to  refer  them  to  three,  the  means  of 
reference,  as  before,  being  called  the  co  ordinates. 


40.  Let  XAY,  XAZ,  and  YAZ,  be  any  three  fixed  planes,  in- 
definite in  extent,  intersecting  each 
other  in  the  Hnes,  AX,  AY  and 
AZ ;  and  let  M  be  any  point  within 
the  angle  formed  by  these  planes. 
Through  this  point,  draw  the  lines 
MP,  MP'  and  MP",  respectively 
parallel  to  the  hnes,  AZ,  AY  and 
AX,  terminating  in  the  planes.  If 
the  distances  MP,  MP'  and  MP",  or 
their  equals,  AR,  AR'  and  AR" 
are  given,  it  is  evident  that  the  position  of  the  point  will  be 
fully  determined,  and  may  be  constructed,  thus :  On  AX  lay  olF 
AR"  =  MP"  ;  on  AY  lay  oflt  AR'  =  MP' ;  through  theii 
extremities  draw  the  lines  R"P  and  R'P  parallel  respectively  to 
AY  and  AX ;  through  their  point  of  intersection,  P,  draw  PM 
parallel  to  AZ,  and  on  it  lay  off  the  given  distance,  MP  ;  the  ex- 
tremity will  be  the  required  point. 

The  planes  X/i.Y,  XAZ  and  YAZ,  are  called  the  co-ordinate 
planes. 

Tlie  first  is  designated  as  the  plane  XY ;  the  second,  as  XZ ; 
and  the  third,  as  YZ. 

Tlie  lines  AX,  AY  and  AZ,  are  the  co-ordinate  axes. 

The  first  is  the  axi^  of  X,  and  the  distances  parallel  to  it  are 
denoted  by  x.  The  second  is  the  axis  of  Y,  and  the  distances 
parallel  to  it  are  denoted  by  y.  The  third  is  the  axis  of  Z,  and 
the  corresponding  distances  are  denoted  by  z.     The  point  A  is  the 


INDETERMINATE    GEOMETRY. 


origin  of  co-ordinates,  and  the  distances  MP,  MP'  and  MP",  are 
the  rectilineal  co-ordinates  of  the  point  M. 


41.  If  the  distances  of  a  point,  from  the  co-ordinate  planes, 
Y'AZ,  XAZ  and  XAY,  are  respectively  denoted  by  a,  h  and  c,  we 
Lave  for  this  point 

X  =i  a  y  =^  h  z  =1  c, 

which  are  the  equations  of  the  2>oint ;  and  when  these  equations 
are  given,  the  point  is  said  to  be  given,  and  may  be  constructed  as 
in  the  j^receding  article. 

The  point  M  is  in  the  first  angle,  that  is,  in  the  angle  to  the 
right  of  YZ,  in  front  of  XZ,  and  above  XY. 

Those  points  which  are  on  the  left  of  the  plane  YZ,-  are  dis- 
tinguished from  those  on  the  right,  by  giving  the  minus  sign  to  x ; 
those  behind  the  plane  XZ,  from  those  in  front,  by  giving  the 
minus  sign  to  y  ;  and  those  below  the  plane  XY  from  those  above, 
by  giving  the  minus  sign  to  z.  Thus,  for  a  point  in  the  second 
angle,  that  is,  in  the  angle  to  the  left  of  YZ,  in  front  of  XZ,  and 
above  XY,  the  equations  are 

2;=—  a  y  =z   h  z  =  c. 

For  a  point  in  the  third  angle,  which  is  immediately  behind  the 
i9cond,  I 

a;=   —  a  '  V  '=■   —   ^  z  ■=  c. 

For  a  point  in  the  fourth  angle,  immediately  behind  the  &st, 

X  =z  a  y  =   —  h  z  =  c. 

For  a  point  in  the  fifth  angle,  under  the  first, 

X  =  a  y  =  &  z  =   --  c. 

For  a  point  in  tlie  sixth  angle,  under  the  second, 


t 


INDETERMINATE    GEOMETRY.  49 

X    =    —    a  y    =    l>  Z    =    —    C. 

For  a  point  in  the  seventh  anyle,  under  the  third, 

X  =^   —   a  7j  =    —   h  z  =1   —  c. 

For  a  point  in  tlie  eighth  angle,  under  the  fourth, 

X  ^=  a  2/=   —   ^  z  =   —  c. 

If  a  point  is  in  the  plane  XY,  the  value  of  z  for  this  point  is  0, 
and  the  equations  of  a  point,  in  this  plane,  are 


X  =  a 


9/  =   b 


0; 


and  there  are  similar  equations  for  points  in  each  of  the  other  co 
ordinate  planes. 

If  a  point  is  on  the  axis  of  X,  the  values  of  y  and  z,  for  this 
point,  are  both  0,  and  the  equations  of  a  point  on  this  axis  are 


y  =   0 


z  =  0; 


and  there  are  similar  equations  for  points  on  each  of  the  other  co- 
ordinate axes. 

The  equations  of  the  origin  of  co-ordinates  are 


X  =   0 


y 


0. 


'  42.  It  is  found  most  convenient,  in  practice,  to  take  the  co- 
ordinate planes  at  right  angles  to  each  other,  and  they  are  always 
considered  to  be  in  this  position,  unless  it  is  otherwise  indicated. 

Let  x\  y\  z',  and  x'\  y", 
2",  be  the  co-ordinates  of  any 
two  points  in  space,  as  M  and 
W.     Then 

x'  =  AT,  y'  =  TP,    z'  =  MP. 

x"  =  AT',  y"  =  T'P',  z"  =  M'F 

Join    P    and  P',    and   draw 
4 


/T^ 


50 


INDETERMINATE    GEOMETRY. 


y  MR    parallel  to    PI''.      Then   from  the  triangle  MRM',   right 
angled  at  R,  we  have 


MM'  =   V^MR*  +  M'R*. 


But,  Art.  (11), 


MR'  =  PP'     =   (x"   -  x'Y  +   (y"   -  yy, 


and 


M'R*  =  (2"  -  z'Y; 
hence,  denoting  the  distance  MM'  by  D,  we  obtain 


D  =    V(x"  -  x'Y  +   (y"  -  y'Y  +  {z"  -  0«; 

or,  the  distance  between  two  points,  iii  space,  is  equal  to  the  square 
root  of  the  sum  of  the  squares  of  the  differences  of  their  co-ordinates. 
If  one  of  the  points,  as  M,  be  placed  at  the  origin,  x',  y'  and  z' 
become  0,  and 


43.  The  position  of  points,  in  space,  may  also  be  determined 
by  referring  them  to  any  other  three 
fixed  objects.  For  instance,  let  A 
be  a  fixed  point,  and  AX  a  fixed 
line  in  the  given  plane  YAX,  and  let 
M  be  any  point  in  space.  If  the  dis- 
tance AM,  and  the  angles  MAP  and 
PAX  are  given,  the  position  of  the 
point  is  known,  and  may  readily  be 
constructed. 

This  method,  in  which  points  are  referred  to  a  fixed  point,  a 
fixed  plane,  and  a  fixed  line  of  the  plane,  is  called  the  system  of 


INDETERMINATE    GEOMETRY. 

2X)lar  co-ordinates  in  space  ;  in  which  the  point  A,  is  the  pole,,  and 
the  distance  AM,  the  radius  vector.  The  three  variable  co-ordi- 
nates are,  the  radius  vector,  the  angle  which  it  makes  with  the 
plane,  and  the  angle  which  its  projection  on  the  plane  makes  with 
the  fixed  line. 

This,  and  the  method  of  rectilineal  co-ordinates,  discussed  in  the 
preceding  article,  form  the  two  principal  systems  of  co-ordinates  in 
space. 


OF    THE    RIGHT    LINE    IN    SPACE. 


44.     Let 


a2  -f  7 


be  the  equation  of  a  right  line,  B'C,  in  the  co-ordinate  plane  XZ, 


and 


y  =^  hz  +  ^ 


the  equation  of  B"C',  in  the  plane 
YZ.  If  through  each  of  these 
lines,  a  plane  be  passed  perpen- 
dicular to  the  planes  XZ  and  YZ 
respectively,  these  planes  will  in- 
tersect in  a  right  line,  BC,  which 
will  thus  be  completely  determined. 


-f  a.. 


•(i: 


/jL   \yii 


The  two  equations 
y  =-  hz  +  (3 


7jr-~x 


•(2), 


taken  together,  may  then  be  regarded  as  the  equations  of  the  right 
line  in  space,  and  when  they  are  given,  the  right  line  will  be  given, 
and  may  be  constructed  by  points.  For,  if  a  value  be  assigned  to 
either  variable,  in  these  equations,  the  values  of  the  other  two  can 
at  once  be  deduced,  and  the  three,  taken  together,  will  be  the  co 
ordinates  of  a  point  of  the  line.  For  instance,  assume  a  value  foi 
z  •=  RP' ;    this,  with  the  corresponding  value  of  x  deduced  from 


0^ 


R/B' 


52  INDETERMINATE    GEOMETRY. 

equation  (1),  will  determine  a  point,  P',  on  the  line  B'C,  through 
-X  which  if  a  perpendicular,  P'M,  be  drawn  to  the  plane  XZ,  it  will 
intersect  the  given  line  in  a  point,  M.  This  same  value  of  z,  with 
the  corresponding  value  of  y,  deduced  from  equation  (2),  will  de- 
termine a  point,  P",  on  B"C',  through  which,  if  a  perpendicular 
Ihj  drav/n  to  YZ,  it  will  intersect  the  line,  in  space,  at  the  same 
point,  M,  since  no  two  points  of  this  line  can  have  the  same  value 
of  r. 

The  two  planes  passing  through  the  line  in  space,  perpendicular 
to  the  co-ordinate  planes,  are  called  the  2^fojectin(jf  2^icLnes  of  the 
line ;  and  the  lines  B^C  and  WC\  in  which  they  intersect  the  co- 
ordinate planes,  are  the  projections  of  the  given  line. 
•  In  equation  (1),  a  represents  the  tangent  of  the  angle  which  the 
projection  of  the  given  line,  on  the  plane  XZ,  makes  with  the  axis 
of  Z,  and 'a'  the  distance  cut  from  the  axis  of  X,  by  the  same  projec- 
tion, Art.  (25). 

In  equation  (2),  b  represents  the  tangent  of  the  angle  which  the 
projection  on  YZ,  makes  with  the  axis  of  Z,  and  /3,  the  distance 
cut  from  the  axis  of  Y. 

If  we  combine  equations  (1)  and  (2),  and  eliminate  the  variable 
x,  we  deduce 

y  _  ^  =  1  (r  -  -«) (3), 

a 

whichj  expressing  the  relation  between  y  and  x  for  points  of  the 
line,  is  evidently  the  equation  of  its  projection  on  the  plane  YX. 


45.  Tlie  principle  that  the  constants  in  the  equation  of  a  line, 
serve  to  determine  it.  Art.  (23),  may  be  well  illustrated  by  sup- 
posing the  four  constants  in  equations  (1)  and  (2)  of  the  preceding 
article,  to  be  given  in  succession.  Thus,  if  a  alone  is  given,  the 
line  is  subjected  to  the  single  condition,  that  its  projection  on  the 
plane  XZ,  shall  make  a  given  angle  with  the  axis  of  Z,  that  is,  it 


^ 


Y'S.gu^-r'^        1    "- 


INDETERMINATE    GEOMETRY. 


may  lie  in  either  one  of  a  sjstem  of  parallel  planes,  perpendicular 
to  XZ,  and  making,  with  the  axis  of  Z,  an  angle,  the  tangent  of 
which  is  a  :  If  a'is  now  given,  the  distance  cut  off  on  the  axis  of  X 
is  known,  and  the  line  may  have  any  position  in  one  of  the  before 
described  planes :  If  6  is  also  given,  the  other  projection  must 
make  a  given  angle  with  the  axis  of  Z,  that  is,  the  line  in  this 
fixed  plane  must  make  an  angle  with  the  axis  of  Z,  the  tangent  of 
which  is  Z»,  or  it  may  occupy  any  one  of  an  infinite  number  of 
parallel  positions  in  this  plane  :  If  /5  is  also  given,  the  line  is  ab- 
solutely fixed. 

If  o^  and  /3  are  0,  the  line  will  pass  through  the  origin  of  co- 
ordinates, and  its  equations  become 

X  =  az,  y  =   hz (1). 

If  in  these, 

0  =   0,         and         h  =   0, 

the  line  will  coincide  ivith  the  axis  of  Z,  and  the  equations  become 

a;  =   0,  ?/  =   0,  2     indeterminate. 

If  the  value  of  z  be  taken  from  the  first  of  equations  (l),  and 
substituted  in  the  second,  we^obtain 

1  h 

z  z=z  —X,  y  =  —X, 

a  a 

for  the  equations  of  the  projections  of  the  right  line,  passing 
through  the  origin,  on  the  planes  ZX  and  YX.     If  in  these, 

-L  =  0  and  A  =  0, 

a  a 

the  line  will  coincide  with  the  axis  of  X,  and  the  equations  of  this 
axis  be 

«  =  0,  y  z=i  0^  X    indeterminate. 


( 


54  INDETERMINATE    GEOMETRY. 

In  a  similar  way,  if  the  line  coincide  with  the  axis  of  Y,  we 
have 

JL  =  0  and  —  =  0, 

b  b 

and  the  equations  of  this  axis  will  be 

z  =   0,  a;  =  0,  y     indeterminate. 


46.  For  the  point  in  which  a  line  pierces  the  plane  XY,  z  must 
be  0.  Substituting  this  value  in  equations  (1)  and  (2)  of  Art. 
(44),  we  have 

X  =  ai  y  =  /3 ; 

hence,  a  and  ^,  taken  together,  are  the  co-ordinates  of  the  points 
in  which  the  right  line  pierces  the  plane  XY. 

In  a  similar  way,  the  co-ordinates  of  the  points  in  which  the 
line  pierces  the  other  co-ordinate  planes,  may  be  determined. 


47.     Let 
X  =  az  +  a. (1),  y  =  bz  +  (3 (2), 

X  =   a'z   +   ?. (3),  y  =   b'z  -h  /3' (4), 

be  the  equations  of  two  right  lines.  If  these  lines  intersect,  or  have 
a  point  in  common,  the  co-ordinates  of  this  point  must  satisfy 
the  equations  at  the  same  time  ;  or  for  this  point,  ar,  y  and  z  must 
be  the  same  in  all  of  the  equations.  Hence,  if  we  combine  these 
equations  and  find  proper  values  for  ar,  y  and  z,  they  will  be  the 
co-ordinates  of  the  common  point.  These  four  equations,  contain- 
ing but  three  unknown  quantities,  can  not  be  satisfied  by  the  same 
set  of  values  if  they  are  independent  of  each  other.  If  the  lines 
intersect,  there  must  then  be  such  a  relation  existing  between  the 


INDETERMINATE    GEOMETRY.  65 

known  quantities  of  the  equations,  as  to  make  one  dependent  upon 
the  other  three,  and  the  equation  which  expresses  this  relation  will 
be  the  equation  of  condition  that  the  lines  shall  intersect. 
Equating  the  second  members  of  (1)  and  (3),  we  deduce 


a  —  a' 
and  in  a  similar  way,  from  (2)  and  (4), 

^'  -  ^ 

Placing  these  values  equal  to  each  other,  we  have 

g^  -  0?    _     (3'  —  (3 
a  —  a'   ~'     h  --  h'' 

or 

(?  -"^)(i  -   h')  =   (^'   -  /3)(a  -  a') (o), 

for  the  equation  of  condition  that  the  lines  shall  intersect. 

This  equation  contains  eight  arbitrary  constants,  any  seven  of 
which  may  be  assumed  at  pleasure,  and  the  remaining  one  thus 
determined,  so  as  to  cause  the  lines  to  intersect. 

Substituting  the  first  of  the  above  values  of  2  in  equation  (1), 
and  the  second  in  equation  (2),  we  find 


aac  —   a' a 
X  =  _-  y  = 


These  values  of  x  and  y,  with  either  value  of  sr,  will  give  a  point 
of  intersection  when  equation  (5)  is  satisfied. 

\i  a  :=  a'  and  h  =  &',  equation  (5)  is  satisfied,  and  the 
values  of  a;,  y  and  z  become  infinite.  The  point  of  interse3tion  ia 
then  at  an  infinite  distance,  that  is,  the  lines  are  parallel, 

a  =  a'  b  =  b' 


56 


INDETERMINATE    GEOMETRY. 


are  then  the  analytical  conditions  that  two  right  lines,  in  space, 
shall  he  parallel.  But  a  =  a'  is  the  condition  that  the  lines 
represented  by  equations  (1)  and  (3)  shall  be  parallel,  Art.  (28), 
and  h  =  h',  the  condition  that  the  lines  represented  by  (2) 
and  (4)  shall  be  parallel.  Hence,  if  two  right  lines,  in  space,  are 
parallel,  their  projections  on  the  same  co-ordinate  plane  will  he 
parallel. 


If  at   the   same   time 


a'     and     (3  =  (3',     the  above 


values  of  z,  x  and  y  become  indeterminate,  as  they  should,  since 
the  two  lines  then  coincide. 


48.  Since  the  angle  included  between  two  right  lines,  in  space, 
is  the  same  as  that  included  between  two  lines  passing  through  a 
common  point  and  parallel  respectively 
to  the  first ;  let  the  lines  AP  and  AP' 
be  drawn  through  the  origin  of  co-ordi- 
nates, parallel  to  any  two  given  lines, 
making  with  each  other  an  angle  de- 
noted by  V.  The  equations  of  AP  and 
AF  will  be 


X  =  az. 


y 


hz: 


X  =  a'z. 


=  h'z 


m  which  a,  h,  a'  and  h',  are  the  same  as  in  the  equations  of  the 
given  lines.  Art.  (44),  and  the  included  angle  is  equal  to  V.  De- 
note the  angles,  made  by  the  first  line  with  the  axes  of  X,  Y  and  Z 
respectively,  by  X',  Y'  and  Z',  and  let  ^X",  Y''  and  Z"  represent 
the  corresponding  angles  made  by  the  second  line. 

Take  any  point,  as  P,  of  the  first  line,  and  denote  its  co-ordinates 
by  x\  y'  and  z',  and  its  distance,  from  A,  by  r',  and  let  x",  y"  and 
z",  be  the  co-ordinates  of  any  point,  as  P',  of  the  second  hue,  and 
r"  its  distance  from  A,  and '  let  D  be  the  distance  PP'.     Then 


from  Trigonometry,  we  have 


CLiAXf 


QJ-^L^ 


/ 


i/ 


INDETERMINATE    GEOMETRY.  57 

2r'r" 
OP 

D3  -   r'2  —  r"2  4.   2rV'  cos  V  =  0 (1), 

in  which,  Art.  (42), 

D«  =  {x'  -  x"y  +  (y'  —  yo^  +  (^''  -  ^"Y (2). 

But  if  from  P,  lines  be  drawn  perpendicular  to  the  axes  of  X, 
Y  and  Z,  respectively,  right  angled  triangles  will  be  formed,  from 
which  we  have 

x'  =  r'  cos  X',         y'  =  r'  cos  Y',         z'  —  r'  cos  Z' (3). 

In  a  similar  way,  we  find 
x"  ?=  r"  cos  X",         y"  =  r"  cos  Y",         z"  =  r"  cos  Z". 

Substituting  these  values  in  equation  (2),  developing  and  ar- 
ranging, we  have 

D8=(cos2X'+cos8Y'4-cos2Z>'2+(cos2X"+cos8Y"+cos2Z")r"2 

--  2  (cos  X'  cos  X"  +  cos  Y'  cos  Y"  +  cos  7J  cos  Z")  r'r", 

and  substituting  this  in  equation  (1),  w^e  have 

(cos^X' + cos^Y' +cos8Z'- 1  )r'2 + (cos«X" + cos^  Y"  +  cos^Z"- 1 )  r"* 

+  2[cos  V-(cosX'cosX"  +  cosY'cosY"+cosZ'cosZ")>'V"  =  0. 

Now  since  the  points  P  and  P'  were  taken  at  pleasure,  and 
sijce  the  angles  V,  X',  X'^,  &c.,  are  entirely  independent  of  the 
distances  r'  and  r",  this  equation  will  be  true  for  any  value  of  r' 
and  r"  ;  it  is  therefore  an  identical  equation,  in  which  the  coefficients 
of  r'2,  r"2^  (fee.,  must  be  separately  equal  to  0  ;  hence 

cos«X'+cos2  Y'+cos«Z'=  1,     cos8X"  +  cos2Y''-f  cos^Z"—  l...(4)j 


58  INDETERMINATE    GEOMETRY. 

COS  V  =  COS  X'  COS  X"  +  COS  Y'  cos  Y"  +  cos  Z'  cos  Z" (5> 

From  equations  (4),  we  see  that,  the  sum  of  the  squares  of  the 
cosines  of  the  angles^  which  any  right  line  makes  with  the  co-ordi- 
nate axes,  is  equal  to  unity,  or  radius  square. 

From  equation  (o),  we  see  that,  the  cosine  of  the  angle  formed 
by  two  right  lines  in  space,  is  equal  to  the  sum  of  the  rectangles  of 
the  cosims  of  the  angles  formed  hy  these  lines  with  the  co-ordinate 
axes. 

49.  Since  the  point  P  is  on  the  line  AP,  its  co-ordinates  x';  y' 
and  z',  must  satisfy  the  equations  of  AP  and  give 

x'   =  az',  y'   =   hz'.... (1). 

Substituting  these  values  of  x'  andj^r^  m  the  ecLuation,  Art.  (42j, 

r'»  =  ic'2  +  y'^  +   2'a 

and  deducing  the  value  of  z',  we  have 


2     = 


Va^   +*52  +   1 
and  this  value  of  2',  in  equations  (1),  gives 

ar'  .  hr' 


Va^  4-^,2+1  •/«=»  +  6«   +   1 

Substituting  these  values  of  a;',  y'  and  2',  in  equations  (3),  of  the 
preceding  article,  we  deduce 

__,  a  h 

cos  X'  =      ,  cos  Y'  = 


-/a*  +  62  +  1  -v^a*  +  i*  +  i' 

1 

cos  Z'    =   •    / :  . 

Va^   +   ja   +    1 

In  a  similar  way,  we  may  deduce 


i 


Y  INDETERMINATE    GEOMETRV.  ^ 


Va'a  +  ^,'2  +  1  y^^a  +  j/a  4.  i  ' 

cos  Z"  =  I 

Va'2  +  6'«  +  1' 

Substituting  these  values  in  equation  (5)  of  the  preceding  arti- 
cle, we  have 

cos  V  =   ±  "^ -^- (3V 

Va2   4.   j2  ^   1  Va'a  +   6'^   +   1 

giving  the  double  sign  as  the  angle  may  be  acute  or  obtuse. 
If  V  =  0,  cos  V  =   1,  hence 

aa'   +   66'   +   1 
1   =     •  -r  ^ 


Va-^  +  68+1   Va'^  +   6'3  +   1 

Squaring  both  members,  transposing  and  reducing,  we  obtain 

(a  -  a'Y  +  (6  -  6')3  +   (ab'  -  a'6)«  =  0, 

and  since  the  first  member  is  the  sum  of  three  positive  terms, 
it  can  not  be  0,  unless  each  term  is  separately  equal  to  0; 
hence 

a  =  a',  6  =  6',  ab'  =  a'6, 

conditions  deduced  in  article  (47),  the  third  evidently  resulting 
from  the  other  two. 

If  V  =  90°,  cos  V  =  0  ;  hence 

aa'  +  66'  +   1   =  0, 

which  is  the  equation  of  condition  that  two  right  lines,  in  spac6^ 
shall  be  perpendicular  to  each  other.  This  equation  being  en- 
tirely different  from,  and  independent  of  equation  (5),  Art.  (47), 
Bhows  that  two  lines  may  be  perpendicular  in  space,  without  in- 
tersecting. 


60  INDETERMINATE    GEOMETRY. 

The  angle,  whicli  the  line  AP  makes  with  the  plane  XY,  is  evi- 
dently the  complement  of  that  which  it  makes  with  the  axis  of  Z, 
and  so  with  the  other  co-ordinate  planes  ;  hence  if  we  denote 
these  angles  by  U,  U'  and  U",  we  have 


sin  U  =  cos  Z',         sin  U'   =: 

cos  y, 

sin  U"  =  cos  X', 

or 

sin  U  -                ^ 

sin  U'  = 

b 

Va^  +  62  +  1 

Va*   +  62   +   1 

sin  U"  -  — 

a 

Va^   +62+1' 

expressions  from  which  the  angles,  made  by  a  given  right  line  with 
the  co-ordinate  planes,  may  be  determined. 

50.     Let 

X  =  az  -\-  "a,  y  =  Z>z  +  /3, 

be  the  general  equations  of  a  right  line,  in  which  <x,  6,  a,  and  /3, 
are  undetermined,  and  let  x',  y\  %'  be  the  co-ordinates  of  a  given 
point.  If  the  line  represented  by  the  above  equations  passes 
through  the  given  point,  its  co-ordinates  must  satisfy  the  equations 
and  give  the  equations  of  condition 

x^  =  az'  ■\-  a,  y'  =   hz'   +  (3. 

If  we  subtract  the  last  equations,  member  by  member,  from  the 
first,  we  shall  introduce  the  conditions  thus  expressed  into  tlie 
first,  eliminate  o^and  /3,  and  obtain 

X-   X'   =   a  {z-  z') (1),        y  -   y   :-_   6  (:;  -  z') (2), 

which  are  therefore,  the  equations  of  a  r'ujht  line  passing  through 
a  given  point  in  space. 

In  these  equations  a  and  h  are  still  undetermined,  as  they 


IXDEThKiMINATE    GEOMETRY.  61 

should  be,  since  an  infinite  number  of  lines  may  pass  through  the 
jriveu  point. 

If  the  line  is  required  to  be  parallel  to  a  given  line,  the  equa- 
tions of  which  are 

X  =  a'z   +   ^,  y  =z   b'z  +  ^', 

a  and  h  will  become  known,  since  we  must  have,  Art.  (47), 

a  =  a',  b  =   b', 

and  by  the  substitution  of  these  values,  the  hue  will  be  fully  de- 
termined. 

Find  the  equation  of  a  right  line,  which  shall  pass  through  the 
point 

x'  =   2,  2/'   =    -   3,  z'  =   1, 

and  be  parallel  to  the  line  of  which  the  equations  are 

X  =^  2z     +3,  y  =    ~  z  +   1. 

51.  If  the  line,  represented  by  equations  (1)  and  (2)  of  the 
preceding  article,  be  subjected  to  the  additional  condition  that  it 
shall  pass  through  the  point  whose  co-ordinates  are  x",  y"  and  z'\ 
these  co-ordinates  must  satisfy  its  equations  and  give  the  equations 
of  condition 

x"  —  x'  =  o{z"  —  z%  y"  —  y  =  b{z"  —  z% 

from  which  we  deduce 

x"  —  x'  y"  —  y' 

^  ^  T^rzrzi '         ^  =  2"  —  ^ ' 

Substituting  these  values  in  the  equations  (1)  and  (2),  we  have 


als< 


62  INDETERMINATE    GEOMETRY. 

which  are  the  equations  of  a  right  line  passing  through  tivo  given 
points  in  space. 

Find  the  equations  of  a  right  line  which  shall  pass  through  the 
two  points 

x'  =  2,     y'  =  0,     2'  =  0;        x"  =:  0,     y"  =  3,     z"  =  —  1. 

52.  Curves,  in  space,  may  be  represented  in  the  same  manner 
as  the  right  line  has  been  represented  in  Art.  (44).  Thus,  if 
through  a  curve,  cylinders  be  passed  whose  elements  are  perpen- 
dicular to  the  co-ordinate  planes,  these  cylinders  will  be  the  pro- 
jecting cylinders  of  the  curve,  and  their  intersections  with  the  co- 
ordinate planes,  the  projections  of  the  curve,  either  two  of  which 
being  given,  by  their  equations,  the  curve  may  be  constructed  by 
points,  ^s  in  Art.  (22). 


53.  The  points  of  intersection  of  two  curves ,  in  space,  may 
so  be  determined  as  in  Art.  (47),  by  combining  their  equations 
But  as  there  will  always  be  four  equations,  involving  but  three  un 
known  quantities,  proper  values  for  the  variables  belonging  to  a 
common  point,  can  not  be  found,  unless  an  equation  of  condition, 
deduced  as  in  that  article,  by  eliminating  x  and  y  and  equating 
the  values  of  z,  shall  be  satisfied. 

To  illustrate  the  intersection  of  two  curves,  let  us  take  the  equa- 
tions 

22*  —  3^:  =   0 (1)  ) 

>•    1st  curve. 
^'^  -  3y  =   0 (2)  ) 

2«   +   3a;2  —  12a;   +   9  =   0 (3)  J 

>•    2nd  curve. 
22   ^   3j^a  __     Qy    ^    0 (4)) 

If  we  combine  equations  (1)  and  (3),  and  deduce  the  values  of « 
and  2,  we  have 


•  L  % 


^    0 


INDETERMINATE    GEOMETRY. 


63 


*  =  2. 


=  dz   Vs 


9 


These  values  of  x  and  z 
are  evidently  the  co-ordi- 
nates of  the  points  M  and 
M',  in  which  the  projections 
of  the  curves  on  the  plane 
XZ  intersect. 

Combining  equations  (2)and  (4),  we  obtain 

y  =  1,        z  =  =fc  Vs ; 

y  =   0,  2  =   rfc  0, 

and  these  are  the  co-ordinates  of  the  points,  A  and  N",  common  to 
the  projections  of  the  curves  on  the  plane  YZ.  The  second  va- 
lues of  z,  in  the  two  cases,  being  unequal,  can  not,  with  the  cor- 
responding values  of  x  and  y,  satisfy  all  four  equations  at  the  same 
time  and  therefore  do  not  belong  to  a  point  common  to  the  two 
curves.  The  first  values  of  z,  viz.  z  =  ±  -y/s^  are  the  same  in 
both  cases  and  therefore  taken  with  ar  =  2,  and  y  =  1,  are  the  co- 
ordinates of  two  points  in  which  the  curves  intersect,  one  of  these 
points  being  above,  and  the  other  the  same  distance  below  the 
plane  XY,  at  P. 

The  same  result  may  be  otherwise  obtained  thus  :  Combine 
equations  (1)  and  (3)  and  eliminate  x,  thus  deducing  an  equation 
involving  z.  Combine  equations  (2)  and  (4)  and  eliminate  y,  thus 
deducing  another  equation  in  z ;  and  since  there  can  be  no  com- 
mon point  unless  these  equations  give  equal  values  for  z,  it  follows 
(the  second  member  of  both  being  0),  that  for  each  equal  value  of 
z  the  first  members  will  have  a  common  divisor  of  the  form  z  ~  a: 
hence,  if  we  seek  the  greatest  common  divisor  of  these  first  mem- 
bers and  place  it  equal  to  0,  the  roots  of  the  resulting  equatiou 


64  INDETERMINATE    GEOMETRY. 

will  give  all  the  values  of  z  whicli  will  satisfy  both  equations. 
Those  which  give  real  values  of  x  in  (1)  and  (3),  and  real  values 
of  y  in  (2)  and  (4),  will  correspond  to  points  of  intersection.  By 
applying  this  process  to  the  above  equations  we  find  for  the  great- 
est common  divisor  z^  —   3,  which  placed  equal  to  0,  gives 

the  same  values  before  found. 

If  only  the  form  of  the  equations  of  two  curves  should  be  given, 
the  constants  which  enter  them  being  arbitrary,  x  and  y  may  be 
eliminated,  as  above,  and  then  such  values  may  often  be  assigned 
to  these  constants,  as  to  give  the  first  members  of  the  resulting 
equations  in  z,  a  common  divisor  of  the  first  or  higher  degree, 
thus  causing  the  two  curves  to  intersect  in  one  or  more  points. 


^-r^  OF    THE   PLANE. 


^ 


54.  The  equation  of  a  surface  is  an  equation  which  expresses 
the  relation  between  the  co-ordinates  of  every  point  of  the  sur- 
face. 

A  plane  surface  may  be  generated,  by  moving  a  straight  line, 
so  as  to  touch  another  straight  line,  and  have  all  of  its  positions 
parallel  to  its  first  position.  The  moving  line  is  called  the  genera- 
trix ;  and  the  line  on  which  it  moves,  or  which  directs  its  motion, 
the  directrix. 


65.     Let 

y  ^'  a'x  -f  h' (1), 

be  the  equation  of  any  right  line,  DB,  in  the  plane  XY,  and  let 
ir  =   az  +  a,  y  =   hz   ■\-  /3 (2), 


INDETERMINATE    GEOMETRY. 


65 


be  tlic  equations  of  a  right  line  in  space,  wliicli  is  to  be  moved  on 
the  hue  DB,  so  as  to  generate  a 
plane.  Since  the  moving  line  must 
always  be  parallel  to  its  first  po- 
sition, a  and  h  will  remain  the  same 
in  all  of  its  positions,  while  a  and  ^ 
will  change,  as  the  line  is  moved 
from  one  position  to  another.  But 
5  and  /3  are  the  oo-ordinates  of  the 
point,  in  which  the  line  pierces  the 
co-ordinate  plane  XY,  Art.  (46),  and  since  this  point  must  be  on 
the  line  DB,  the  values  of  a  and  /3,  deduced  from  equations  (2), 
must,  in  all  positions  of  the  generatrix,  satisfy  equation  (l),  when 
substituted  for  the  variables.  •    The  values,  thus  deduced,  are      y 

CL  =^  X  —   az^  /3   =   y  —    Jr, 

and  these,  substituted  for  x  and  y  in  equation  (1),  give 

y  —   hz  =   a'{x  —   az)   -{-   h' (3),  -^ ^f/^" 

which  expresses  a  relation  between  the  co-ordinates  of  the  different 
points  of  the  generatrix,  in  all  of  its  positions  ;  it  is,  therefore,  the 
equation  of  a  plane.  If  this  equation  be  solved  with  reference  to 
z,  and  the  coefficients  of  x  and  y  be  placed  equal  to  c  and  d,  re- 
spectively, and  the  absolute  term  equal  to  ^,  we  have  ^^-^ 

z  —  ex   -^    dy   -\-  g (4),    ^_^^^ 

a  form  analogous  to  that  of  the  right  line,  Art.  (24). 

Since  this  equation  contains  three  variables,  either  two  may  be 
assumed  at  pleasure  and  the  corresponding  value  of  the  third  de- 
duced;  the  three,  taken  together,'  will  be  the  co-ordinates  of  a 
point  of  the  plane,  which  may  be  constructed  as  in  Art.  (40),  and 
as  any  number  of  its  points  may  be  determined  in  the  same  way,  the 
plane  will  evidently  be  given  when  the  constants  which  enter  its 
equation  are  known. 
5 


4 


\ 


;.t 


P" 


QQ  INDETERMINATE    GEOMETRY. 

And,  in  general,  any  surface  will  be  given,  analytically,  Mben        fl 
the  form  of  its  equation  and  the  constants  which   enter  it  are 
known.  ™ 

b(j.  The  intersection  of  a  plane  with  either  co-ordinate  plane 
is  called  a  trace  of  the  plane. 

For  every  point  of  the  plane,  which  lies  in  the  co-ordinate  plane 
XZ,  y  must  be  equal  to  0.  Substituting  this  value  for  y,  in 
equation  (4)  of  the  preceding  article,  we  obtain 

z    —    ex    +    (J (1),  >, 

in  which  x  and  y  can  only  belong  to  points  of  the  plane  lying  in 
the  plane  XZ.  This  is  then  the  equation  of  the  trace,  BC,  on  the 
plane  XZ. 

In  the  same  wa}',  for  all  points  of  the  plane,  in  YZ,  x  must  be 
equal  to  0  ;  Avhence 

*  =  ^y  +  9 (2), 

is  the  equation  of  the  trace,  DC,  on  the  plane  YZ. 
By  making     z  =  0,      we  obtain 

ex  -\-  dy  ■\-  g  =  0, 

for  the  equation  of  the  trace,  BD,  on  the  plane  XY. 

For  all  points  in  the  axis  of  Z,  x  and  y  must  be  equal  to  0. 
Substituting  these  values  for  x  and  y  in  equation  (4),  we  find 


which  is  the  distance  AC,  cut  off  by  the  plane  on  the  axis  of  Z. 

Tn  a  similar  way,  we  find  the  distances  cut  off  on  the  axes  of  X 
andY 

ar  =    -   A  =   AB,  y  =    _   Z_  =  AD. 

c  d 


^roy4-*.H+^ 


INDETERMINATE    GEOMETRY.  67 

If  ^  =  0,  these  distances  become  0,  the  plane  will  pass 
through  the  origin,  and  its  equation  become 

z  —  ex  ■\-  dy, 

without  an  absolute  term,  as  it  should  be,  since  the  co-ordinates  of 
the  origin  w^ill  then  satisfy  the  equation. 

If  c  =  0,  the  distance  AB  becomes  infinity,  and  the  plane 
is  parallel  to  the  axia  of  3^,  or  perpendicular  to  the  co-ordinate 
plane  YZ,  and  its  equation  becomes 

z  =  c?y  -f  y, 

the  same  as  that  of  the  trace  on  ZY.  It  should  be  remarked, 
however,  that  for  the  plane,  x  may  have  any  value,  or  is  indeter- 
minate, since  its  coefficient  c  is  0  ;  while  for  the  trace,  x  must  be 
equal  to  0,  as  we  have  seen. 

If  cZ  =  0,  the  distance  AD  becomes  infinity,  and  the  equa- 
tion of  the  plane  perpendicular  to  XZ, 

z  =1  ex  '\-  g,  y     indeterminate. 

In  the  same  way,  if  equation  (3),  Art.  {oo)^  had  been  solved 
with  reference  to  y  or  rr,  it  might  be  shown  that  the  equation  of  a 
plane  perpendicular  to  XY,  would  be  the  same  as  that  of  its  trace^ 
z  being  indeterminate. 


h*l.     Every  equation  of  the  first  degree  between  three  variables, 
will  be  a  particular  case  of  the  general  equation 

A.r  +  By  +  Cr  +  D  =   0, 
and  this,  when  solved  with  reference  to  z,  gives 

A  B  D 

«  =   —   — x  —   — ?/  _  — , 

c         iy       0 

an  equation  of  the  same  nature  and  form  as 


68  INDETERMINATE    GEOMEITIY. 

z  ==  cx  +  dt/  -^  g (1), 

and  will  therefore  represent  a  magnitude  of  the  same  kind ;  tliat 
is,  every  equation  of  the  first  degree  between  three  variables  is  the 
equation  of  a  plane,  and  when  solved  with  reference  to  z,  will  ap- 
pear under  the  form  (1). 

58.     Let 

X  =^   az   +  ^,  y   Tn   bz  ■{-  (3 (1), 

be  the  equations  of  a  right  line,  and 

z  —  ex  +  dy  -}-  g (2), 

the  equation  of  a  plane.  Those  values  of  x,  y  and  z  which,  when 
taken  together,  will  satisfy  these  three  equations  at  the  same  time, 
must  be  the  co-ordinates  of  a  Point,  common  to  the  line  and 
plane.  Therefore,  by  combining  the  equations  and  deducing  the 
values  of  ar,  y  and  z,  we  shall  obtain  the  co-ordinates  of  the  point 
in  which  the  line  pierces  the  plane.  Substituting  the  values  of  x 
and  y,  from  equations  (1),  in  equation  (2),  we  find 

ac  +  ^d  '\-  g, 
It  ^^  .   .» — i — .  ■  -■ — * 

1   —   ac  —   fee? 

and  by  the  substitution  of  this  value  of  z  in  equations  (1),  we  may 
deduce  the  corresponding  values  of  x  and  y.     If 

1   —  ac  —  6i  ==   0, 

tlie  values  of  z,  x  and  y  will  become  infinite,  the  point  in  which 
the  line  pierces  the  plane  will  be  at  an  infinite  distance,  and  the 
line  will  he  parallel  to  the  plane^.  The  last  equation  is  then  the 
analytical  condition  that  a  right  line  shall  be  parallel  to  a  plane ; 
or,  that  a  right  line,  having  one  point  in  a  plane,  shall  he  wholly 
in  the  plane. 


j 


INDETERMINATE    GEOMETRY. 


69 


In  tlie  same  way,  the  points  in  which  any  line,  in  space,  pierces 
a  surface  may  be  found ;  since  the  two  equations  of  the  line,  with 

'the  equation  of  the  surface,  will  always  give  three  equations,  by 
the  combination  of  which,  values  of  the  three  variables  may  be 
deduced  which  will  satisfy  the  equations  at  the  same  time.  The 
number  of  sets  of  real  values  thus  found  will  indicate  the  num- 

*  bor  of  common  points. 


59.     Let 


=   ex  -\-  dy   4-   <7, 


be  the  equation  of  a  plane,  and  suppose  any  straight  line  to  be 
drawn  perpendicular  to  the  plane.  If  through  the  point  where 
the  plane  cuts  the  axis  of  Z,  a  line  be  drawn  parallel  to  the  given 
line,  its  equations  will  be  of  the  form 


X  =^  az  •\-  a, 

in  which  a  and  h  are  the  same  as 
in  the  equations  of  the  given  line, 
Art.  (47).  Since  this  second  linei 
is  also  perpendicular  to  the  plane, 
it  must  be  perpendicular  to  the 
traces,  BC  and  DC,  which  are  two 
lines  of  the  plane  passing  through 
its  foot.  The  equations  of  the 
trace  BC,  Art.  {5^)^  Tciay  be  put 
under  the  form 


y   —   bz   ■{■   8, 


X  •=  — z  — 
c 


y  =  0.2 , 


since  the  projection  of  BC,  on  the  plane  YZ,  coincides  with  the 
axis  of  Z. 

The  general  equation  of  condition  that  the  right  line  shall  b« 
perpendicular  to  the  trace  is.  Art.  (49), 


70  INDETERMINATE    GEOMETRY. 

1   +  aa'   +  hh'  =   0 (1), 

in  whicli,  from  the  above  equationsj^he  trace^  we  must  have 

a'  =  JL  6'  =  0. 

c 

Substituting  these  values  in  equation  (1),  we  obtain 

1    +   -^  =   0 (2),  or         a  =   —  c 

c 

for  the  condition  that  the  line  shall  be  perpendicular  to  the  trace. 
In  a  similar  way,  for  the  trace  DC,  we  have 


h'  =  L 
d 

a'   =   0, 

and  these,  in  equation  (1),  give 

1    +  -^  =   0 (3), 

or 

6  =   -  (f . 


—  c  b  =   —  d 


are  then  the  analytical  conditions  that  a  straight  line  shall  be  per- 
pendi*^ular  to  a  plane. 

Condition  (2)  proves  also  that  the  projection  CM  is  perpen- 
dicular to  the  trace  BC,  Art. '(28) ;  and  condition  (3)  proves  that 
the  projection  CM'  is  perpendicular  to  DC.  Hence,  if  a  right  line 
is  perjtendicular  to  a  plane ^  its  projections  are  2>erpe7idicular  to  the 
traces  of  the  plane^  respectivehj. 

60.     Let  a:',  y\  z\  be  the  co-ordinates  of  a  given  point,  and 

z  =  cz  -\-  dy  ■\-  g (1), 

tlic  equation  of  a  given  plane.    The  equations  of  a  right  line  pass- 
ing through  the  given  point  will  be.  Art.  (50), 

r  —  y   =   a(2r  -  z')  y  -   y'   =   b{z  -   z') (2). 


INDETERMINATE    GEOMETRY.  1l 

If  tbis  line  is  required  to  be  perpendicular  to  the  plane,  we 
must  have,  by  the  preceding  article, 

a  —  —  c,  b  =  —  d. 

Substituting  these  values  in  equations  (1),  we  have 

X  -  x   =  -  c{z  -  s'),         ,j  ^  y'  =  -  d(z  -  z') (3), 

for  the  equations  of  a  right  line  passing  through  a  given  point 
and  perpendicular  to  the  plane. 

The  point,  in  which  this  perpendicular  pierces  the  plane,  mav 
be  found,  as  in  Art.  (58),  by  combining  equations  (3)  with  equa- 
tion (1) ;  and  the  distance  between  this  and  the  given  point,  or 
the  length  of  the  perpendicular,  by  means  of  the  formula  of 
Art.  (42). 

Find  the  equations  of  a  straight  line  passing  through  a  point 
whose  co-ordinates  are 

X'  =    -   2,  y'  =   1,  z'   =   3, 

and  perpendicular  to  the  plane  whose  equation  ir> 
2x  —   37/  +  4z  -\-   1    =0. 

Find  also  the  point  in  which  the  line  pierces  the  plane,  and  the 
length  of  the  perpendicular.      ^  . 

61.  The  angle,  made  by  a  straight  line  with  a  plane,  is  the  same 
as  the  angle  included  between  the  line  and  its  projection  on  the 
plane.  Therefore,  if  through  any  point  of  the  line  a  perpendicular 
be  drawn  to  the  plane,  this  perpendicular,  a  portion  of  the  line  and 
its  projection  on  the  plane,  will  form  a  right  angled  triangle,  of 
which  the  angle  at  the  base  will  be  the  angle  made  by  the  line 
and  plane,  and  the  angle  at  the  vertex,  its  complement. 

Denote  the  first  angle  by  A,  and  the  angle  formed  by  the  given 
lint5  and  the  perpendicular  by  V.      Then,  the  line  being  repre* 


72  INDETERMINATE    GEOMETRY. 

sented  by  equations  (1)  and  (2),  Art.  (44),  and  tlie  plane  by 
equation  (4),  Art.  (55),  the  perpendicular  will  be  represented  by 
equations  (o)  of  the  preceding  article,  and  by  substituting  —  c  for 
a',  and  --   d  for  h'  in  the  formula  (3),  of  Art.  (49,)  we  have 


cos 

V  == 

± 

1    — 

ac    — 

ou 

—  sin 

A, 

Vl    +  a 

'  + 

T^  vT 

+ 

c2 

+  d'^ 

from  ^ 

^vhich 

we 

determine 

the 

sine  of 

A, 

and 

thence  the 

angle 

itself. 

K 

1    - 

ac  - 

-  hd  = 

=   0, 

the  angle  becomes  0,  and  the  line  is  parallel  to  the  plane,  a  con- 
dition before  determined,  Art.  (58). 


^ 


62.     Let 

z  =z  ex  -{-  d9/  +  g (1), 

z  =  c'x^-  d'y  -f  9' (2), 


be  the  equations  of  two  planes.  Those  values  of  x,  y  and  z  which 
will  satisfy  both  of  these  equations,  at  the  same  time,  must  belong 
to  points  common  to  the  two  planes.  If  then  we  combine  these 
equations,  ar,  y  and  z  in  the  result  can  only  belong  to  the  line  of 
intersection  ;  and  if  one  of  the  variables,  as  z,  be  eliminated,  we 
have 

(,   _   c')x  -\-  {d  ~  d')y   Jr  0   -  9'  =   0 (3), 

which  must  be  the  equation  of  the  projection  of  this  hne  of  inter- 
section on  the  plane  XY.  In  the  same  way,  if  the  equations  be 
combined  and  x  be  eliminated,  the  result  will  be  the  equation  of 
the  projection-  of  the  hne  of  intersection  on  the  plane  YZ.  Two 
projections  being  thus  determined,  the  line  will  be  known. 

If  such  a  relation  exists  between  c,  c',  fi  and  d\  that  no  values 


INDETERMINATE    GEOMETRY.  73 

of  X  and  y  "will  satisfy  equation  (3),  the  planes  can  not  intersect, 
but  must  be  parallel.  This  can  only  be  the  case  when  c  =^  c' 
and     d  =  d'^   as  we  shall  then  have 

^  -  y  =  0, 

which  can  not  be  if  the  planes  are  different ;  hence, 

c  =^  c'^  d  =  d\ 

are  the  anahjtical  conditions  that  two  planes  shall  he  parallel. 

By  referring  to  the  equations  of  the  traces  of  these  planes,  we 
f?ee  that     c  ^=  c'     is  the  condition  that  the  traces  on  the  plane 
ZX  shall  be  parallel,  Art.  (28),  and  that     d  =  d'     is  the  con 
dition  that  the  traces  on  the  plane  ZY  shall  be  parallel ;  hence, 
if  two  planes  are  parallel,  their  traces  are  parallel. 

If  the  plane  represented  by  equation  (1)  is  parallel  to  the  co- 
ordinate plane  XY,  its  traces  on  XZ  and  YZ  must  be  parallel, 
respectively,  to  the  axes  of  X  and  Y ;  hence,  by  a  reference  to  the 
equations  of  these  traces.  Art.  {^Q)<,  we  see  that 

c  =   0,  c?  =   0, 

and  that  equation  (1)  reduces  to 

z  ^=  (J,  X  and  y  indeterminate, 

for  the  equation  of  a  plane  parallel  to  the  co-ordinate  plane  XY. 
If    ^  =  0,     also,  we  have 

«  =   0,  X  and  y  indeterminate, 

for  the  equation  of  the  co-ordinate  plane  XY. 

If  the  plane  represented  by  (1)  is  parallel  to  the  co-ordinate 
plane  YZ,  its  traces  on  XZ  and  XY  must  be  parallel  to  the  axes 
of  Z  and  Y  which  requires 

±  =  0,  -1=0. 

C  C 


74  INDETERMINATE    GEOMETRY. 

These  values,  substituted  in  equation  (1),  placed  under  the  fonn 

Ida 

X  =  —z y  -  ^, 

c  c  c 

give 

X  =   —    ^         or  re  =  /i,         y  and  z  indeterminate, 


for  the  equation  of  a  plane  parallel  to  YZ,  and  at  a  distance  from 

it  equal  to     —  —  =  /i  • 
c 

If    ^  =  0,    also,  we  have 

a;  =   0  y  and  z  indeterminate, 

for  the  equation  of  the  plane  YZ  ;  and  similar  equations  may  be 
found  for  a  plane  parallel  to  XZ,  and  for  the  plane  XZ  itself. 

The  preceding  method  of  finding  the  intersection  of  two  planes 
is  applicable  to  any  surfaces  whatever.  Thus  :  Combine  the  equa- 
tions of  the  surfaces,  and  eliminate  one  of  the  variables,  the  result 
will  be  the  equation  of  the  projection  of  the  intersection  on  the 
plane  of  the  other  two  variables.  Combine  the  equations  again 
and  eliminate  another  variable,  the  result  will  be  the  equation  of 
the  projection  on  another  plane,  and  the  intersection  will  be  thus 
determined. 

Find  the  intersection  of  the  two  planes  whose  equations  are 

2x  —  3y  +  2r  =  0, 

a;  +  2y  —  3^  -f   1   =  0. 

63.  If  through  any  point,  within  the  angle  included  by  two 
planes,  a  line  be  drawn  perpendicular  to  each  plane,  the  angle  in- 
cluded by  one  of  these  lines  and  the  prolongation  of  the  other, 
will  be  equal  to  the  angle  included  by  the  planes.      Let  the  equa- 


INDETERMINATE    GEOMETRY.  76 

tions  of  the  planes  be  the  same  as  in  the  preceding  article^  then 
the  equations  of  the  perpendiculars  will  be,  Art.  (60), 

a;  —  a;'  =  —  c  (s  —  z'\  y  —  y'  ^  —  d  {z  —  z'), 

a:  —  ar*  =  —  c'  (z  —  2'),  y  —  y'  z=  —  d' (z  —  z% 

If  we  denote  the  angle  which  these  lines  make,  by  A,  and  then 
substitute  —  c  and  —  c'  for  a  and  a\  and  —  d  and  —  d'  for  b 
and  b',  in  formula  (3),  Art.  (49),  we  have 

A          ^                     1   +  cc'   +  dd'  ,,, 

cos  A  =   ±  — (1), 

Vl   +  c2  +  d*  Vl   +  c'*  4-  d'^ 

from  which  we  deduce  the  value  of  cos  A,  and  thence  of  A  itself, 
which  will  express  tlie  number  of  degrees,  &c.,  contained  in  the 
angle  of  the  planes. 

If  the  two  planes  are  parallel,  we  have  A  =  0,  cos  A  =  1. 
By  substituting  this  value  of  cos  A,  clearing  of  denominators,  &c., 
as  in  Art.  (49),  we  may  deduce  the  same  equations  of  condition  as 
in  the  preceding  article. 

If  the  two  planes  are  perpendicular  to  each  other,  we  must  havo 
A  =   90°,     cos  A  =   0,     which  requires 

1  +  cc'   +  dd'  =   0, 

the  equation  of  condition  that  two  planes  shall  be  perpendicular  to 
each  other. 

If  the  first  plane  coincides  with  the  plane  X'^  we  have,  from 
the  preceding  article, 

c  —  0  d  =  0, 

and  cos  A  reduces  to 


cosX''  =  ^ 

Vl   4-  c'3   f  d'» 


76  INDETERMINATE    GEOMETRY. 

for  the  cosine  of  the  angle  made  by  tlie  second  plane  with  the 
co-ordinate  plane  XY. 

Kthe  same  plane  coincides  with  the  plane  YZ,  we  tiave 

i   =   0,  ^  =   0, 

C  C 

and  tkese  values  substituted  in  equation  (1),  first  placing  it  under 
the  form, 

_   4-   c'  -^ d' 

cos  A  = 


/l  ,         d^      , 

V^   +   1    +   -   -v/l    +  c'«   +  c^' 

reduce  it  to 


cos 


Y"  = 


Vl    +   c'a   -f    c/'« 


for  the  cosine  of  the  angle  made  by  the  second  plane  with  the 
plane  YZ. 

If  the  plane  coincides  with  XZ,  we  have 

i  =  0,  i  =  0, 

d  d 

and  equation  (1)  may  be  reduced  to 

d' 


cos 


Z"  = 


Vl   +  c2  +  d'» 


In  the  same  way,  if  the  second  plane  be  made  to  coincide,  in 
succession,  with  each  co-ordinate  plane,  we  may  deduce  for  the 
angles  X',  Y'  and  Z',  made  by  the  first  plane  with  the  co-ordinate 
planes 


cos  X'  =  — z==^==^  1        cos  Y'   r=   —  , 

Vl    +  c"  +  d*  -y/i  +  c3  +  d^ 


INDErKUMJNATl':    GEOMETRY,  77 

d 


COS 


Z'  = 


Vl     +    C*     -f    rf2 


If  both  members  of  these  three  equations  be  squared  and  the 
results  added,  member  to  member,  we  find 

cos^X'  +  cos^Y'   +  cos^Z'  =  1. 

If  the  values  of  cos  X'  and  cos  X''  be  multiplied  togethe?,  also 
cos  Y'  and  cos  Y",  cos  Z'  and  cos  Z"  and  the  three  products 
added,  we  obtain 

cos  X'  cos  X"  +  cos  Y'  cos  Y"  +  cos  Z'  cos  Z"  =  cos  A, 

an  expression  for  the  cosine  of  the  angle  formed  by  two  planes,  in 
terms  of  the  cosines  of  the  angles  made  by  the  planes  with  the 
co-ordinate  planes. 


64.  Let  ir',  ?/',  z',  be  the  co-ordinates  of  a  given  point,  and 

z  =   ex   -{•  dy   '\-  g (1), 

the  general  equation  of  a  plane,  in  which  c,  d  and  g  are  arbitrary 
constants.  If  the  given  point  is  in  this  plane  its  co-ordinates  must 
satisfy  the  equation  and  give  the  equation  of  condition, 

z'   =   ex'   -f  dy'   +  g. 

Subtracting  this  equation,  member  by  member,  from  (1),  we  in- 
troduce the  condition  into  that  equation  and  obtain, 

z  ^  z'  =  c[x  —  x')   -t-  d{y  —  y'), 

for  the  equation  of  a  plane  passing  through  a  given  point,  in 
which  c  and  d  are  still  arbitrary. 


Go.  If  the  plane,  reprcK'iited  by  equation  (1)  of  the  preceding 


t  S  INDETERMINATE    GEOMETRY. 

article,  be  required  to  contain  the  three  given  pointt  x\  y\  z\  x'\ 
y",  s",  and  x'",  y'"^  z"\  these  co-ordinates,  when  substituted  in 
succession  for  the  variables,  must  satisfy  the  equation  and  give 
the  three  equations  of  condition, 

%'  =  ex'  +  dy'  +  ^, 
z"  =  ex''  +  dy"  +  <7, 
z'"  =   ex'"  +  dy'"  +  ff. 

From  these  three  equations,  the  values  of  the  three  constants  c, 
d  and  ff  may  be  determined,  and  substituted  in  equation  (1). 
The  result  will  be  the  equation  of  a  plane  passing  through  three 
given  points. 

Find  the  equation  of  a  plane  passing  through  the  three  points, 

X'    =  1,        y'    =  0,        z'    =   -  d; 
x"   =   2,         y"  =   1,         z"   =   1] 
x'"  =   0,         y'"  =   2,         z'"  =   0. 


v' 


v^ 


\ 


^* 


TRANSFORMATION    OF    CO-ORDINATES. 


^  66.  In  developing  and  discussing  the  properties  of  lines  and 

surfaces,  it  is  often  of  great  advantage  to  change  the  reference  of 
their  points  from  one  system  of  co-ordinate  axes  or  planes  to  ano- 
ther. The  system  from  which  the  change  is  made  is  called  the 
primitive  system  ;  the  one  to  which  it  is  made  is  the  new  system  ; 
and  changing  the  reference  of  points,  from  one  system  of  co-ordi- 
nate axes  or  planes  to  another,  is  called  the  transformation  of  co- 


if a  line  or  surface  be  given  by  its  equation,  and  it  be  required 
to  change  the  reference  of  its  points  to  a  new  system  of  co-ordinate 
axes  or  planes  ;  it  is  only  necessary  to  deduce  values  for  the  primi- 
tive co-ordinates  in  terms  of  the  new,  and  to  substitute  these  values 


INDETERMINATE    GEOMETRY.  <9 

for  the  variables  in  the  given  equation.  The  result,  expressing  a 
relation  between  the  new  co-ordinates  of  the  points,  will  of  course 
be  the  equation  of  the  line  referred  to  the  new  system. 

From  the  nature  of  this  operation,  it  is  evident  that  no  change 
whatever  takes  place,  either  in  the  nature  or  extent  of  the  line  or 
surface. 


f 


67.  Let  AX  and  AY  be  any  set  of  co-ordinate  axes,  and  AX' 
and  AY'  any  other  set  having  the  same  origin.  Denote  the  angle 
included  between  AX  and  AY  by  /3, 
and  let  a  and  a^  denote  the  angles 
made  by  AX'  and  AY',  respectively, 
with  AX,  Let  AP  =  x  and 
MP  =  y  be  the  co-ordinates  of  any 
point,  as  M,  when  referred  to  the  first 
set,  and  let  AP'  =  x'  and 
MP'  =  y'  be  the  co-ordinates  of  the  same  point  referred  to  the 
second  set.  Through  P'  draw  P'R  parallel  to  AX  and  P'S  paral- 
lel to  AY. 

In  the  triangle  ASP',  the  angle 

AP'S  =  P'AY  =  ^  —  a,  sin  ASF_=jin^YAX  =  sin^,_ 

and  since  the  sides  are  as  the  sines  of  their  opposite  angles,  we 
have  the  two  proportions, 

AS  :  AP'  :  :  sin  (/3  -  a)  :  sin  ASF  or  sin  /S,  "^ 

P'S  :  AP'  :  :  sin  oT  :  sin /3 

whence 

AS  —  ^'  ^^^^  (^  "~  ^  P'S  —  ^'^^^  ^ 

sin  ^  '  Bin  /} 

In  the  triangle  P'RM, 


80  INDETERMINATE    GEOMETRY. 

y'         FMR  =  YAY'  =  /3  -^',  MP'R  =  Y'AX  =  a! 

^  ^  MRP'  =  P'SA, 

id  we  liave  the  proportions 

PR  :  FM  :  :  sin  (/3  -  a')  :  sin  /S, 
MR>  :  P'M  :  :  sin  ?  :  sin  /3 ; 


whence 

P'R  =  y'  ^^^  ^^  ""  ^') 
sin  jS 

We  have  also 

AP  =  AS  +  P% 


MR  = 


sin  /3 


MP  =.  P'S  +  MR. 


Substituting,  in  these  equations,  the  vahies  above  deduced,  we 
have 


X   = 


y  = 


x'  sin  (/3  —  'a)   +  y'  sin  (/3 

sin  /3 

rr'  sin  a  +  y'  sin  a' 
sin  /3 


in  which  the  values  of  the  primitive  co-ordinates  are  expressed  in 
terms  of  the  new  and  constants  ;  and  these  are  the  formulas,  for 
passing  from  any  system  of  rectilineal  co-ordinates  to  another 
having  the  same  origin. 

Tf  the  new  origin  is  different  from  the  primitive,  at  A',  for  in- 
stance, it  is  evident  that  we  have 
¥  simply  to  add  to  the  above  values, 
a'  and  &',  the  co-ordinates  of  the  new 
origin  referred  to  the  primitive  sys- 
tem.    We  thus  obtain 


a;  =  a'   -I- 


x'  sin  (]S   —   a)   -f  y'  sin  (/3   —  a') 
sin  /3 


f 


INDETERMINATE    GEOMErKV.  81 

x'  sin  a   +  y'  sin  a' 


V  + 


sin  /5 


general  formulas  for  2^oissing  from  one  system  of  rectilineal  co-ordi- 
nates  to  any  other,  in  the  same  plane. 

If  the  new  axes  of  co-ordinates  are  required  to  be  parallel  to 
tlie  primitive,  we  have 

CH  a  =   0,         a^  =  /3,         sin  a  =   0,         sin  a'  =  sin  /3, 

and  the  above  formulas  reduce  to 

X   =   a'    -{-   x',  y   =   h'    +   y> (2), 

formulas  for  passing  from  any  set  of  co-ordinate  axes  to  a  parallel 
set,  in  the  same  plane. 

If  the  primitive  axes  are  perpendicular  to  each  other,  we  have 

^13  =  90°,     sin  /3  =   1,     sin  (/3  —  a)   =  cos  a, 
sin  ((3  —   a')  =  cos  a', 

and  formulas   (1),  reduce  to 

X  =  a'  ■}•  x'  cos  a,  -\-  y'  cos  a' 

(3). 

y  =  h'   +  ^'  sin  a  +  y'  sin  a' 

formulas  for  passing  from  a  system  of  rectangular  co-ordinate  axes 
to  an  oblique  system,  in  the  same  plane. 

If  the  primitive  axes  are  perpendicular  to  each  other,  and  also 
the  new,  we  have 

/3  =   90°,     sin  (3  =   1,     a'  =   90°   -f  a,     sin  a'   =  cos  a, 

sin  {(3  —  a)  =  cos  a,         sin  (/3  —  a')  =  sin(—  a)=  —  sin  a, 

and  formulas  (1)  reduce  to  M 


\k 


X  =  a'   +  a;'  cos  a  —  y'  sin  a  /V 

w, 

y  =z  h'   ^  x'  sm  a,   -\-  y'  cos  a 
6 


82  INDCTERMINATE    GEOMETRY. 

formulas  for  passing  from  a  srjstem  of  rectangular  co-ordinate  axes 
to  another  system^  also  rectangular^  in  the  same  plane.  H 

If  the  new  axes,  only,  are  perpendicular  to  each  other,  we  have  \ 

V- 

a'   =   90°   +   a,         sin  a'  =  cos  a,         cos  a'  =   —  sin  a,         K 


and  formulas  (1)  reduce  to 

(/3  -  « 

sin  ^ 


X  =  a'   ^  ^^^inj^-  a)  -  y'  cos  (^  -  a) 


.(5), 


7,     .     a:'  sin  a  +  y'  cos  a 
y  =  5'   4-  Z_^^ 

sm  p 

formulas /or  passing  from  a  system  of  oblique  co-ordinate  axes  to  a 
rectangular  system,  in  the  same  plane. 

If  the  new  origin  be  the  same  as  the  primitive,  a'  and  h'  in  each 
of  the  above  formulas  will  be  equal  to  0. 

68.     We  may  illustrate  the  use  of  the  formulas  of  the  preceding 
article  by  the  following 

Examples. 

1.  Let 

x^  +  y""  -=  R' (1), 

be  the  equation  of  a  circle  referred  to  its  centre  and  rectangular 
co-ordinate  axes.  Art.  (35),  and  let  it  be  proposed  to  change  the 
reference  to  a  parallel  set  having  the  origin  at  the  point  C. 

27  The  co-ordinates  of  the  new  origin  will 

be 

a'  =   —  R,  h'  =  0, 

B 

and  these  values,  in  formulas  (2),  reduce 

them  to 


r 

^ 


INDETERMINATE    GEOMETllY.  83 

X  =  —  R  +  x',  y  =  y'. 

Substituting  these  last  values  for  x  and  y  in  equation  (1),  and 
reducing,  we  obtain 

2/'2  =    2Kx'  -  x"^, 

an  equation  before  found  in  Art.  (38). 
2.  Let 

y  z=  ax  +  h (2), 

be  the  equation  of  the  right  line  A'B,  referred  to  the  rectangular 

axes  AX  and  AY,  and  let  it  be  proposed 

to  find  the  equation  of  the  same   line     X^ 

referred  to  the  axes  A'X'  and  A'Y',  also 

at  right  angles,  the  axis  of  X'  making 

an  angle  of  45°  with  the  axis  of  X  and 

having  the  new  origin  at  A',  the  point  -^  -^ 

where  the  given  line  cuts  the  axis  of  Y.     The  general  formulas  to 

be  used  in  this  case  are  formulas  (4),  in  which 

^  a'  =   0,  6''  =   6,  sin  a  =  cos  a. 

These  values  reduce  the  formulas  to 

X  =   i^x'  —   y')  cos  a,  y  =   6   +   (a:'   +  y')  cos  a, 

and  substituting  these  values  for  x  and  y,  in  equation  (2),  we  have 

h  -\-  {x'   +  y')  cos  a  =  a(x'   —  y')  cos  a   +  &, 

or  reducing, 

y'  =  X . 

a  +   1 

€9.     Let  AX  and  AY  be  a  set  of  rectangular  co-ordinate  axes, 


INDETERMINATE    GEOMETRY. 

and  M  any  point  referred  to  tliera 
by  the  co-ordinates  AR  =  a*,  and 
MR  =  y\  and  let  P  be  the  pole,  and 
PS  the  fixed  Hne,  to  which  the  point  is 
referred  by  the  radius  vector  PM  =  r 
and  the  angle     MPS  =  v,    Art.  (18). 

SPT  =  a. 

In  the  right  angled  triangle,  MTP,  we  have 

PT  =  r  cos  {y  +  a),  MT  =  /•  sin  (y  -f-  a). 

Substituting  the  above  values  in  the  equations 

AR  =  AO  +  PT,  MR  =  OP  4-  MT, 

we  have 

X  =  a'  •\-  r  cos  (v  -f  a),         ^^  =   ^'   +  '' sin  (v  +  a) (1), 

which  are  general  formulas,  for  passing  from  a  system  of  rect- 
angular co-ordinates  to  a  system  of  polar  co-ordinateSj  in  the  same 
plane. 
The  fixed  line  is  generally  taken  parallel  to  the  axis  of  X,  in 
J^   which  case     a  =  0,     and  formulas  (1)  reduce  to 

X  =^  a'  -{•  r  cos  v,  y  =   Z»'  +  »*  sin  v (2). 

*  If  the  pole  is  at  the  origin,  we  have     a'  =  0,     h'  =■  0. 
From  the  second  of  equations  (1),  we  deduce 


sin  (v  +  a) 


in  which,  if    y  >  ^',     y  —  h'    is  positive,  the  point  M  is  above 
^    the  line  PT,  and    sin  (v  +  a)  also  positive ;  hence,  the  value  of 
r  will  be  essentially  positive. 


INDETERMINATE    GEOMETRY. 


86 


y 


If  y  <i  h',  y  —  h'  is  negative,  the  point  M  is  below  FT, 
sin  {v  -f-  a)     also  negative,  and  the  value  of  r  positive. 

The  value  of  the  radius  vector  is  therefore  always  positive. 
Hence,  if  in  discussing  the  equation  of  a  line  referred  to  a  fixed 
point  and  fixed  right  line,  usually  called  the  polar  equation  of  the 
line,  a  negative  value  of  the  radius  vector  is  found,  it  must  be  re- 
jected, as  there  can  be  no  corresponding  point. 


a 


70.     To  illustrate  the  principles  of  the  preceding  article,  let  it 

be  proposed  to  determine  and  discuss  the  polar   equation   of  the 
circle.     Its  equation  referred  to  the  y 

rectangular  axes  AX  and  AY,  is 


X?   -f  f 


R2. 


.(1). 


Suppose  the  fixed  line  PS,  fi-om 
which  the  angle  v  is  estimated,  is 
parallel  to  the  axis  of  X,  we  must 
then  use  formulas  (2).     Squaring  the  values  of  x  and  y,  we  have 

x^  =  a'^   -j-  2a'r  cos  v  -\-  r^  cos**  i', 
y«  =   6'2  4-  26V  sin  v  -{-  r^  sin**  v. 
Substituting  these  values  in  equation  (1),  recollecting  that 
sin*  v  -f  cos*  V  =   1, 
and  reducing,  we  obtain 

r*  -f  2{a'  cos  V  +  b'  sin  v)r  +  a'*  +  6'«  —  R«  =  0 (2), 

for  the  general  polar  equation  of  the  circle. 

By  attributing  particular  values  to  a'  and  b\  the  pole  may  be 
pl^ed  at  any  point  in  the  plane  of  the  circle. 
TpJQJie  pole  be  placed  at  C,  we  must  have 


66 


INDETERMINATE    GEOMETRY. 


a'    =     —    R,  1/    =z    0, 

and  these,  in  equation  (2),  give 

r^  —  2R  cos  vr  =  0. 

^ 

This  equation  gives  two  values  of  r, 


r  =  0, 


r  =  2  R  cos  V. 


These  two  values  of  r  represent  the  distances  from  the  pole  to 
the  points  in  which  the  radius  vector,  making  any  angle  v,  cuts 
the  circle.  Since  the  pole  is  on  the  curve,  one  of  these  values 
is  necessarily  0,  whatever  be  the  angle  v.  The  second  may  then 
represent  any  radius  vector  as  CM. 


If  in  this  second  value     v  =  0, 


we  have     cos  v  =   1,     and 
r  =  2R  =  CB, 


which  gives  the  point  B.  As  v  in- 
creases, cos  V  will  remain  positive  until 
V  =  90°,  in  which  case  cos  v  =  0, 
r  becomes  0,  and  the  radius  vector 
takes  the  position  CM'  tangent  to  the  circle  at  C.  As  v  increases 
beyond  90°,  its  cosine  becomes  negative,  the  value  of  r  is  negative 
and  gives  no  point  of  the  curve,  until  v  becomes  equal  to  270°, 
when  cos  V  =  0  and  r  =  0,  taking  the  position  CM'". 
As  V  increases  beyond  270°,  its  cosine  is  positive,  r  is  positive  and 
gives  j  oints  of  the  curve  until  v  =  360°,  when  we  again  have 
r  =  CB. 

From  this  we  see  that  as  v  increases  from  0  to  90°,  we  obtain 
all  the  points  in  the  semi-circumference  BDC,  that  no  points  of 
the  curve  are  on  the  left  of  the  line  M'M'",  and  that  as  v  increases 
trom  270°  to  360°,  we  obtain  all  the  points  in  the  other  semi- 
circumference. 

The  second  value  of  r  is  readily  verified,  since  in  the  right 
angled  triangle  CMB,  we  have 


INDETERMINATE    GEOMETRY.  87 

CM  =  CB  COS  BCM  or  r  =  2R  cos  r. 

If  the  pole  is  placed  at  B,  we  have 

a'  =  R,  6'  =  0, 

and  equation  (2)  gives  the  two  values 

r  =  0,  r  =   —   2R  cos  v. 

The  second  value  of  r  will  be  negative  for  all  valu(  ?  of  v  less 
than  90°  or  greater  than  270°,  and  positive  for  all  values  from 
90°  to  270°. 

Tf  the  pole  is  placed  at  the  centre,  we  have 

a'  ==  0,  V    =  0, 

and  equation  (2)  reduces  to 

r  =  R, 
V  being  indeterminate^  since  its  coefficient  is  equal  to  0. 


71.  By  reflecting  upon  the  discussion  contained  in  the  three 
preceding  articles,  we  see  that  two  classes  of  propositions  may 
arise  in  the  transformation  of  co-ordinates. 

First;  when  it  is  proposed  to  change  the  reference  from  a 
given  set  of  co-ordinate  axes  to  another  set,  the  exact  position  of 
which  is  known.  In  this  case  the  constants  which  enter  the  values 
of  the  primitive  co-ordinates  are  given. 

Second  ;  wh.en  it  is  proposed  to  change  from  a  given  set  to  ano- 
ther, the  position  of  whicli  is  to  be  determined,  so  that  the  result- 
ing equation  shall  assume  a  certain  form,  or  the  new  set  fulfil 
certain  conditions.  In  this  case,  the  constants  above  referred  to 
are  arbitrary,  and  by  assigning  values  to  them,  as  many  reasona- 
ble conditions  may  be  introduced  as  there  are  such  constants,  and 
the  position  of  the  new  co-ordinate  axes  thus  determined. 


68 


INDETERMINATE    GEOMETRY. 


72.     Let  AX,  AY  and  AZ,  be  three  co-ordinate  axes  at  right 

angles  to  each  other,  and  AX', 
AY'  and  AZ',  three  oblique  axes 
having  the  same  origin.  De- 
note the  angles  made  by  the 
new  axis  AX'  with  the  three 
primitive  axes  of  X,  Y  and  Z, 
respectively,  by  X,  Y  and  Z, 
those  made  by  the  axis  AY' 
with  the  same,  by  X',  Y'  and  Z', 
and  those  made  by  AZ',  by  X",  Y"  and  Z". 

Let  M  be  any  point,  in  space,  referred  to  the  primitive  planes 
by  the  co-ordinates  x,  y  and  z.  Through  this  point  draw  the  line 
MP  parallel  to  AZ',  until  it  pierces  the  new  plane  X'Y',  in  the 
point  P ;  through  this  last  point,  draw  PR  parallel  to  AY',  until  it 
intersects  the  new  axis  of  X',  in  R ;  then 


AR 


PR  =  y',  MP  =  z'. 


are  the  co-ordinates  of  the  point  M  referred  to  the  oblique  co-ordi- 
nate planes.  Through  the  points  M,  P  and  R,  pass  planes  paral- 
lel to  the  plane  XY,  intersecting  the  axis  of  Z  in  M',  P'  and  R'. 
AM'  is  equal  to  z,  and  the  lines  AR,  RP  and  PM,  are  the  hypothe- 
nuses  of  right  angled  triangles,  the  bases  of  which  are  AR',  RR" 
and  PP",  and  the  angles  at  the  bases,  Z,  Z'  and  Z".  From  these 
triangles  we  have 

AR'  =  AR  cos  Z,     RR"  =  RP  cos  Z',    PP"  =  MP  cos  Z" 

Substituting  these  values  for  their  equals  in  the  equation 

AM'  =  AR'  +  R'P'  +  P'M', 
and  for  AM',  AR,  RP  and  MP,  their  values,  ve  have, 
2  =  rr'  cos  Z  -f-  y'  cos  Z'  -\-  z    cos  Z". 


INDETERMINATE    GEOMETRY.  89 

In  a  similar  way,  by  drawing  lines  through  the  point  M  respec- 
tively parallel  to  the  new  axes  of  X'  and  Y',  we  may  deduce 

a;  =  a;'  cos  X  +  y'  cos  X'   +  z'  cos  X'', 
y  =  ic'  cos  Y  +  y'  cos  Y'   +  2'  cos  Y". 

These  three  equations  taken  together  express  the  values  of  the 
primitive  co-ordinates  in  terms  of  the  new,  and  are  the  formulas 
for  changing  the  reference  of  points  from  a  set  of  co-ordinate 
planes  at  right  angles,  to  another  set  oblique  to  each  other,  having 
the  same  origin. 

If  the  origin  be  also  changed  to  a  point  whose  co-ordinates  are 
a,  h  and  c,  these  formulas  become 

a;  =  a  -f  a;'  cos  X  +  y'  cos  X'   -j-  ^'  cos  X", 

y  =  6  -f-  a:'  cos  Y  +  y  cos  Y'   +  z'  cos  Y", (1). 

z   =  c  +  ic'  cos  Z    +  y'  COS  Z'  -f  z'  cos  Z", 

In  these  formulas  there  are  twelve  constants ;  but  since  the 
angles  X,  Y,  Z,  &c.,  made  by  each  of  the  new  axes  with  the  prim- 
itive, must  fulfil  the  condition  expressed  in  equation  (4),  Art.  (48), 
thus  forming  three  equations  of  condition,  we  can,  by  means  of 
these  constants,  introduce  only  nine  independent  conditions. 

If  the  new  axes  are  also  perpendicular  to  each  other,  we  shall 
have  the  cosines  of  the  angles,  included  between  each  set  of  two, 
equal  to  0.  Placing  the  expressions  for  these  cosines.  Art.  (48), 
each  equal  to  0,  we  have  three  more  equations  of  condition  exist- 
ing between  the  arbitrary  constants. 

If  the  new  axes  are  parallel  to  the  primitive,  we  have 

X  =  0,  Y'  =  0,  Z"  =  0, 

and  each  of  the  other  angles  equal  to  90°,  hence  the  above  formu- 
las reduce  to 

a;  =  a  +  a;',        y  =  6  +  y',         «  =  c  +  2' (2),  ^ 


+ 


90 


INDETERMINATE    GEOMETRY. 


wliich  are  the  formulas  for  passing  from  a  set  of  planes  at  right 
angles,  to  a  parallel  set. 


13.  Let  M  be  any  point,  in  space,  referred  to  the  three 
rectangular  co-ordinate  planes,  by 
the  co-ordinates 

AR  =  X,    AR'  =  y,     MP  =  r, 

and  to  the  fixed  plane  XY,  the  line 
AX  and  the  point  A,  by  the  polar 
co-ordinates.  Art.  (43), 

AM  =  r,  MAP  =  V,  RAP  =  u. 

The  right  angled  triangles  ARP  and  MPA,  give 

AR  =  AP  cos  u,       RP  =  AP  sin  w, 
MP  =  r  sin  v,  AP  —    r  cos  v. 

Substituting  the  value  of  AP,  the  first  three  equations  give 

X  ^:^  r  cos  V  cos  u,     y  =z  r  cos  v  sin  w,     z  =  r  sin  v (1), 

which  are  formulas  for  passing  from  a  system  of  rectangular  co- 
ordinates to  a  system  of  polar  co-ordinates,  in  space. 
From  the  last  of  the  above  equations,  we  have 


sin  V 


and  since  z  and  the  sin  v  will  always  have  the  same  sign,  the  radius 
vector  will  always  be  positive. 

The  equations  of  the  radius  vector  in  any  one  of  its  positions, 
will  be  of  the  form.  Art.  (45), 


X  =  az, 


y  =  ^^ (2), 


whence 


INDETERMINATE    GEOMETRY.  91 


a 


f.  h^t 


Substituting  tlie  values  of  a;,  y  and  «,  taken  from  form  alas  (1), 
we  have 

a  =  cot  V  cos  w,  ft  =  cot  V  sin  w, 

and  these,  in  equations  (2),  give 

a;  =  cot  V  cos  uz^  y  =  cot  v  sin  wr, 

which  will  be  given,  when  v  and  u  are  known. 


A 


OF    THE    CYLINDER. 


74.  A  cylindrical  surface  or  cylinder ^  may  be  generated  by 
moving  a  straight  line,  so  as  to  touch  a  given  curve  and  have  all 
of  its  positions  parallel  to  its  first  position. 

The  moving  line  is  called  the  generatrix  ;  and  the  given  curve 
the  directrix  of  the  cylinder. 

The  different  positions  of  the  generatrix  are  called  elements  of 
the  surface. 

The  curve  of  intersection  of  the  cylinder,  by  any  plalie,  may  be 
regarded  as  the  base  of  the  cylinder ;  and  when  the  elements  are 
perpendicular  to  the  base,  the  surface  is  a  right  cylinder. 


75.  If  the  directrix  of  the  cylinder  is  a  plane  curve,  its  plane 
may  be  taken  for  the  co-ordinate  plane  KY,  and  its  equation  may 
be  represented,  generally,  by 

/(^,y)  =  0 (1),  ^ 

which  is  read,  a  function  of  x  and  y  equal  to  zero  ;  the  first  mem- 
ber being  a  symbol  to  indicate  an  expression  containing  x,  y  and 


92 


INDETERMINATE    GEOMETRY. 


constants  ;  or  that  x  and  y  are  so  connected  that  one  can  not  vary 
without  the  other. 
Let 


az  +  a, 


hz  +  /3, 


be  the  equations  of  a  right  Hne  which  is  to  be  moved  so  as  to  gen- 
erate the  surface.  Since  the  different  positions  of  this  generatrix 
are  parallel,  a  and  h  remain  constant,  while,  as  the  line  is  moved 

from  one  position  to  another,  a 
and  /3  must  change.  But  a  and 
^  are  the  co-ordinates  of  the  point 
in  which  the  generatrix  pierces 
the  plane  XY,  Art.  (46),  and 
since  this  point  must  be  on  the 
directrix  CD,  the  values  of  a  and 
ft  when  substituted  for  x  and  y,  must  satisfy  equation  (1).  These 
values  are 

a  =  a;  —  az,  (3  =  y  —  bz, 

and  when  substituted  in  equation  (1),  give 

f(x  —  az,     y  —   hz)  =  0, 

an  equation  expressing  the  relation  between  the  co-ordinates  of 
ihe  different  points  of  the  generatrix  in  all  of  its  positions.  It  is, 
therefore,  the  general  equation  of  a  cylinder,  of  which  the  directrix 
may  be  regarded  as  the  base. 

In  order  then,  to  obtain  the  particular  equation  of  a  cylinder, 
whose  directrix  is  given,  we  have  simply  to  substitute,  for  x  and  y 
in  the  equation  of  the  directrix,  the  expressions 


az. 


y  —   ^-. 


76.     If  the  directrix  is  a  circle,  whose  equation  is 
x^  ^  y^  =  R«, 


INDETERMIXATE    GEOMETRY.  9f3 

the  origin  being  at  the  centre,  we  have,  by  making  the  substitu- 
tions above  referred  to, 

(^  «  azy  +  (y  -   hzY  =  R« (1), 

the  equation  of  an  oblique  cylinder  with  a  circular  base. 

If  this  cylinder  be  intersected  by  a  plane  parallel  to  XY,  the 
equation  of  which,  Art.  (62),  is 

z  z=z  g^  X  and  y  indeterminate, 

we  have,  by  combining  the  equations.  Art.  (62), 

{x  -  agY  +  (y  --  hyf  =  W, 

for  the  projection  of  the  curve  of  intersection  on  XY.  But  this  is 
evidently  the  equation  of  a  circle,  whose  radius  is  R,  Art.  (34), 
and  therefore  equal  to  the  base.  But  since  this  intersection  is 
parallel  to  the  plane  XY,  its  projection  is  evidently  equal  to  the 
line  itself.  We  therefore  conclude,  that  if  a  cylinder,  with  a  cir^ 
cular  base,  be  intersected  by  a  plane  parallel  to  the  base,  the  inter- 
section will  be  a  circle  equal  to  the  base. 

If  a  and  h  are  equal  to  0,  the  generatrix  becomes  parallel  to  the 
axis  of  Z,  or  perpendicular  to  the  base,  the  cylinder  becomes  right, 
and  equation  (1)  reduces  to 

^^   +   y^   ==   R2, 

the  same  as  the  equation  of  the  base,  z  being  indeterminate. 


OF    THE    CONE. 


lt£.   V,         «r^ 


VV.  A  conical  surface^  or  cone,  may  be  generated  by  moving  a 
straight  line,  so  as,  continually,  to  pass  through  a  fixed  point  and 
touch  a  given  curve. 


f*- 


94 


i 


INDETERMINATE    GEOMETRY. 


I'he  fixed  poi  ot  is  the  vertex  of  the  cone,  and  the  parts  of  th« 
surface  separated  by  the  vertex  are 
called  nappes. 

The  intersection  of  the  cone  by  any 
plane  may  be  regarded  as   its  base. 
If  the  rectilinear  elements  all  make 
the  same    angle    with    a   right   line  ^f 
passing  tli rough  the  vertex,  the  cone     . 
is  a  right  cone,  and  the  right  line  is    / 
its  axis. 


18.  If  the  directrix  of  the  cone  is  a  plane  curve,  its  plane  may 
be  taken  as  the  co-ordinate  plane  XY,  and  its  equation  be  repre- 
sented as  in  article  (75),  by 


A^,y) 


.(1). 


If  x',  y'  and  z'  are  the  co-ordinates  of  the  fixed  point,  or  vertex, 
the  equations  of  the  generatrix  will  be.  Art.  (50), 


x  —  x'  ■=  a{z  —  z'),  y  —  y'  z=  h[z  ^  z'). 


■(2), 


in  which  a  and  b  change  as  the  generatrix  is  moved  from  one 
position  to  another.     These  equations  may  be  put  under  the  form, 


X  =  az  -\-   (x'   —  az'\ 
m  which  the  absolute  terms, 
x'  —  az', 


y    =    6z    +    (y'    -    bz% 


y'   -    bz\ 


>6yS^^ 


are  the  co-ordinates  of  the  point,  in  which  the  line  pierces  the 
plane  XY,  Art.  (46),  and  since  this  point  is  on  the  directrix,  what- 
ever be  the  position  of  the  generatrix,  these  values,  when  substi- 
tuted for  X  and  y  in  equation  (1),  must  satisfy  it,  and  give 


/(^'   -  az\     y'  -  bz')  =   D. 


INDETERMINATE    GEOMETRY.  96 

Substituting  in  this  equation,  the  values  of  a  and  6,  in  terms  of 
z,  y  and  «,  deduced  from  equations  (2), 

X  —  x'  7       y  —  y' 

a  =     -,  b  =  1 ^ 

z   —  z'  z  —   z' 

we  have  after  reduction, 

/ff^i^^f,       2^1ZL4f^=0 (3), 

\       Z  —    Z  Z    —    Z'    J 

an  equation  expressing  the  relation  between  ar,  y  and  z,  for  all 
positions  of  the  generatrix.  It  is,  therefore,  the  general  eqiudion  of 
a  cone,  of  which  the  directrix  may  be  regarded  as  the  base. 

In  order  then  to  obtain  the  particular  equation  of  a  cone,  whose 
directrix  is  given,  we  have  simply  to  substitute  for  x  and  y,  in  the 
equation  of  the  directrix,  the  expressions, 

x'z  —  z'ar                     y'z  —  z'y 
_ ,  -_  . 

z  —  z'  z  -^  z 


7 9.     If  the  directrix  is  a  circle,  whose  equation  is 
a:«  +  2/»  =  R2, 
we  have,  by  making  the  substitutions  above  referred  to, 


/x'%  —  z'x\^         fy'z  —  z'y\^  _ 


R^ 


or 


{x'%  -  z'xY  +   (y'2  —  xhjf  =  R2  (r  -  Z'Y (1), 

for  the  equation  of  an  oblique  cone  with  a  circular  base. 

If  this  cone  be  intersected  by  a  plane  parallel  to  XY,  the  equa- 
tion of  which.  Art.  (62),  is 


96  INDETERMINATE    GEOMETRY. 

:  z  =  ffy  X  and  y  indeterminate, 

we  have,  by  combining  the  equations, 

{^'9  -  ^'xf  +  {y'g  -  z'yY  =  R^  (.7  -  z% 

for  the  projection  of  the  curve  of  intersection  on  XY.  By  di- 
viding both  members  by  2;'*,  this  equation  may  be  put  under  the 
form 


(a  - .)%  (^  _  ,j  .  5! 


which  is  the  equation  of  a  circle,  the  co-ordinates  of  whose  centre 

are    —    and  —  ,    and  the  radius,  the  square  root  of  the  second 

z'  z' 

member.  Art.  (34).  This  projection  being  equal  to  the  curve  itself, 
we  conclude,  that  if  a  cone,  with  a  circular  base,  be  intersected  by 
a  plane  parallel  to  the  base,  the  intersection  will  he  a  circle.  The 
radius  of  this  circle  will  decrease  as  g  increases,  until  g  =  z', 
when  the  radius  becomes  0  and  the  equation  takes  the  form 

{x'  -  xy  +  {y'   -  yf  =  0, 
which  can  only  be  satisfied.  Art.  (49),  by  making 

X  =  x\  y  =  y', 

and  the  circle  becomes  a  point. 


80.     If 

X'  =  0,  y  =  0,  z'  =  A, 

the  vertex  of  the  cone  is  on  the  axis  of  Z,  at  a  distance,  from  the 
origin,  represented  by  h ;  the  cone  becomes  right,  and  equation 
(1),  of  the  preceding  article,  becomes 


or 


INDETERMINATE    GEOMETRY. 

{x^  +  y')  h^  =  ^'  {z  -  h)\ 


(^*  +  y«)^^  =  (z  -  hy 


d1 


.(1). 


\ 

/ 

^'\ 

/j"-^" 

"I^^^A 

o^'  ^^ 

A/V     iK 

-X 

If  the  angle,  made  by  the  elements  of  the  cone  with  the  plane 
of  the  base,  be  denoted  by  v,  we  have  in  the  right  angled  triangle 
VAB', 


tangAB'V^^, 


or 


tanjr  V  = 


R 


and  equation  (1)  becomes 

{x^  +  y*)  tangS  v  =  {z  —  hy (2), 

for  the  equation  of  a  right  cone  with  a  circular  base. 


yi 


8 1 .    Through  the  axis  of  Y,  in  the  figure  of  the  preceding  article, 
let  a  plane  be  passed  intersecting  the  cone.    This  plane  being  per- 
pendicular to  the  plane  XZ,  its  equation  will  be  the  same  as  that  of 
its  trace  on  XZ,  y  being  indeterminate,  Art.  (56).     Let  the  angle,. 
7 


i 


\  «.  f^  ^  ^ 


,3) 


-¥- 


"99C  INDETERMINATE    GEOMETRY. 

which  this  plane  makes  with  XY,  be  denoted  by  ?/,  the  equation 
of  its  trace,  AR,  will  be,  Art.  (24), 


The  equations  of  the  curve  of  intersection  of  the  plane  and  cone 
may  now  be  found,  as  in  article  (62).  But  as  the  different  curves, 
obtained  by  changing  the  position  of  the  cutting  plane,  form  a 
class  possessing  very  remarkable  properties,  the  discussion  of  which 
is  much  simplified  by  referring  the  intersection  to  lines  in  its  own 
plane,  the  latter  method  is  chosen. 

Let  us  then  take  the  right  lines  AX'  and  AY,  as  a  new  system 
of  rectangular  co-ordinate  axes,  and  let  us  estimate  the  positive 
values  of  x'  from  A  to  X',  and  the  positive  values  of  y'  from  A 
to  Y. 

Let  M  be  any  point  of  the  curve  of  intersection.  Its  co-ordi- 
nates, referred  to  the  primitive  planes,  are 

a:  =  AP,  y  =  MR,  z  =  RP, 

and  referred  to  the  new  axes,  AX'  and  AY, 

-  x'  =  AR,  y'  =  MR. 

Fi-ora  the  right  angled  triangle  APR,  we  have 

AP  =  AR  cos  u,  RP  =  AR  sin  w, 

or 

X  =:   —  x'  COS  Uy  X  =  —  x'  sin  u. 


We  have  also 


1/   =   !/ 


If  these  values  of  x,  y  and  z  be  substituted  in  equation  (2)  of 
the  preceding  article,  the  result  expressing  a  relation  between  x'^ 
and  y'  for  points  common  to  the  plane  and  cone  only,  wjll  be  the 
equation  of  the  intersection.     Making  the  substitution,  we  obJain 

(x'^  cos'  u  -{•  y'^)  tang'^  v  =   (—  a;'  sin  u  —  A)*, 


INLETERMINATli    GEOMETRY.  99 

or  performing  the  operation  indicated  in  the  second  member,  and 
transposing, 

l/"^  tang'*  V  =  x''^  sin*  u  —  x'^  cos;  u  tang^  v  -\-  2x'h  sin  ii  +  A*, 

or  recollecting  that 

sin**  u  =  cos^u  tang*  w,  '-<^*-*^ 

and  omittinoj  the  dashes  of  the  variables, 

//*  tang*  V  =  x^  cos*  u  (tang*  u  —  tang*  v)  -1-  2xk  sin  t«  +  /a*...(1), 

for  tlie  equation  of  ike  line  of  intersection  of  a  plane  and  right  cone 
with  a  circular  base. 

In  this  equation,  h  may  now  be  regarded  as  the  distance  from 
the  vertex  of  the  cone  to  the  point  in  which  the  plane  cuts  the 


C-«-a^ 


82.  If  in  the  above  equation,  v  remaining  the  same^  all  values 
be  assigned  to  u  from  0  to  90°,  and  all  values  to  A,  from  0  to  in- 
tinity,  it  will  represent,  in  succession,  every  line  which  it  is  possi- 
ble to  cut,  from  a  given  right  cone  with  a  circular  base,  by  a  plane. 

There  are  three  distinct  cases. 

First,  when 


^S 


or 


tang  u 


In  this  case,  the  cutting  plane  makes  the  same  angle  with  the 
base  that  the  elements  do,  or  is  parallel  to  \  I  / 

one  of  the  elements,  and  since  ^'"' 

tang*  u  =  tang*  v,  / 

the  coefficient  of  a:*  becomes  0,  the  equation         />'^''"'/"  v^--. 
reduces  to  (LJi-S.-±-j — -- 

y*  tang*  v  =  2xh  sin  u  +  A*, 


and  the  curve  represented  by  it  is  called  a  Parabola. 


^«/^V  ^  J-^ 


100  INDETERMINATE    GEOMETRT. 

If  in  this  equation,  A  =  0,  the  cutting  plane  passes  tlirough 
the  vertex,  and  the  equation  reduces  to 

y^  tang*  v  =  0, 

which  can  only  be  satisfied  by  making 

y  =  0, 

which,  since  x  is  indeterminate,  is  the  equation  of  the  axis  of  X, 
Art.  (21).     A  right  line  is  there/ore  regarded  as  a  particular  case 
of  the  parabola. 
Second,  when 

u  <C  Vj        or        tang  u  <  tang  v. 

In  this  case,  the  cutting  plane  makes  a  less  angle  with  the  base 
than  the  elements  do,  or  is  parallel  to  none  of  the  elements,  see 
figure  of  Art.  (80)  ;  and  since, 

tang'  u  <  tang'  v, 

the  coefficient  of  x^  is  essentially  negative  and  the  curve  represent- 
ed by  the  equation  is  called  an  Ellipse. 

If  in  this  case  w  =  0,  the  cutting  plane  is  parallel  to  the 
base, 

cos  2«  =  1,  sin  ««  =  0,         tang  ?«  =  0, 

and  the  equation  reduces  to 

y^  tang=»  u  =    —  a;»  tang'  v  -f  A*,  ^ 

or  dividing  by  tang'  v  and  transposing 

A' 


2/'   +  a;'  = 


tang*  V 


which  is  the  equation  of  a  circle,  Art.  (35). 

If    A  =  0,     u  being  still  less  than  v,  the  plane  passes  through 
the  vertex,  and  the  equation  reduces  to 


INDETERMINATE    GEOMETRY. 

y*  tang'  v  =  a;'  cos*  u  (tang'*  u  —  tang*  v), 


•101^ 


the  first  member  of  which  is  essentially  positive  and  the  second 
negative ;  it  can  therefore  be  satisfied  for  no  values  of  x  and  y, 
except 


:c  =   0, 


y  =   0, 


which  are  the  equations  of  the  oi-igin  of  co-ordinates,  Art.   (16). 
A  circle  and  point  are  therefore  regarded  as  particular  cases  of  the 
ellipse. 
:^  Third^  when 


u  y  V, 


or 


tang  u   >  tang  v. 


In  this  case,  the  cutting  plane  makes  a  greater  angle  with  the 
base  than  the  elements  do,  or  is  parallel  to  two  of  the  elements, 
viz.  those  cut  from  the  cone  by  passing  a 
plane  through  the  vertex  parallel  to  the 
cutting  plane,  and  since 

tang*  u  >  tang*  v, 

the  coefficient  of  x^  is  essentially  positive, 
and  the  curve  represented  by  the  equation 
is  called  an  Hyperbola. 

If  in  this  case,     ^  =  0,     the  equation 
reduces  to 

y*  tang2  V  =  x"^  cos'  w  (tang'  u  —  tang'  v), 

both  members  of  which  are    essentially  positive.     Dividing  by 
tang*  V,  and  placing 


=  >*, 


we  obtain 


cos'  u  (tang*  u  —  tang'  v) 
tang*  V 

y*  =  r«x«,  y  =   ±  r:p, 


-102;  INDETERMINATE    GEOMETRY. 

wliicli,  evidently,  represents  two  right  lines  intersecting  at  the  ori- 
gin of  co-ordinates,  Art.  (24),  the  equations  of  which  are 

1/  =    ■\-  rx,  y  =  —  rx. 

Two  right  lines,  tvhich  intersect,  are  therefore  regarded  as  a  par* 
ticular  case  of  the  hyperbola. 


83.  Resuming  equation  (1),  Art.  (81),  dividing  by  tang*  v, 
and  denoting  the  co-efficient  of  x^  by  r*,  as  above,  we  have 

y*  —  r^x*    -\-    2x  -i- (1). 

tang*v  .        tang*  v 

Now  let  us  transfer  the  reference  of  the  points  of  the  curve  to  a 
set  of  parallel  co-ordinate  axes,  having  their  origin  at  D,  the  point 
in  which  the  curve  is  cut  by  the  axis  of  X,  [see  figure  of  Art. 
(80)].     Formulas  (2),  of  Art.  (67),  become  for  this  case, 

X  ~  a  +  x',  y  =  y\ 

a  representing  the  distance  —  AD,     and  6'being  equal  to  0. 
Substituting  these  values  in  equation  (1),  we  have 


/'2    _    r2. 


,o   ,    «  /A  sin  w     .      3  \ 

;'2  ^  2  ( \-  r^a  \ 

\^tang*  V  J 


9  „     .     rt  A  sin  M  .         h^ 


tang*  V  tang*  v 

The  origin  of  co-ordinates  being  on  the  curve,  the  absolute 

tcnn 

,  -         2/i  sin  u  A*        -u 

tang*  V  tang*  v 


Note.    It  should  be  observed,  that  by  placing  the  absolute  term 


tar  sr^  v  tang^  v 


INDETERMINATE    GEOMETRY.  108 

must  be  equal  to  0,  Art.  (38),  and  the  equation,  after  omitting 
the  dashes  and  placing 

^  sin  w      ,      „ 
tang'*  V 

reduces  to 

ya  _  ,.2^8  ^   22)x (2), 

a  general  equation,  which  may  represent  either  of  the  above 
named  curves ;  the  parabola  when  r*  =  0,  ike  ellipse  when 
r*  <   0,     and  the  hyperbola  when     7-^  >•   0. 


OF     THE     PARABOLA. 

84.  If  ?"*  =  0,  equation  (2)  of  the  preceding  article,  be- 
comes 

2/'  =   22)X (1). 

This  equation  being  of  the  second  degree,  the  line  represented 
by  it  is  of  the  second  order.  Art.  (33),  and  2^9  being  the  only  con- 

W3  have  an  equation  of  the  second  degree,  and  therefore  two  values  of  a, 
which  will  fulfil  the  required  condition.  Solving  the  equation,  substitu- 
ting the  value  of  r^  and  reducing,  we  find 

_  h  (tang  u  +  tang  v) 

~         cos  u(ta.ng^u  —  tang^i;) 

In  the  parabola,  u  being  equal  to  v,  the  fiirst  value  reduces  to  _  ,  and  the 

second,  to  infinity,  but  by  striking  out  the  common  factor, 
tang  u  —  tang  v,  the  first  value  becomes  finite  and  negative,  as  it  should 
be  to  give  the  point  D. 

In  the  ellipse,  the  first  value  is  negative,  the  other  positive,  the  negative 
value  being  used. 

In  the  hyperbola,  both  values  are  negative,  the  one  which  is  numeri- 
cally the  least  being  used. 


104 


INDETERMINATE    GEOMETRY. 


slant,  the  line  is  given  when  2pis  given,  Art.  (23).  This  con* 
slant  is  called  the  parameter  of  the  parabola,  and  since  from  equa* 
lion  (1),  AYO  may  deduce  the  proportion 

X    \    y    '.',    y    :    2p, 

we  say,  the  parameter  is  a  third  proportional  to  the  abscissa  and  or- 
dinate of  any  point  of  the  curve. 


85.     If  equation  (1),  of  the  preceding  article,  be  solved  with 
reference  to  y,  we  have 

y  =   zt   V2px, 

For  every  positive  value  of  x,  there  will  be  two  corresponding 
real  values  of  y  ;  hence,  the  curve  is  con- 
tinuous and  extends  from  the  orig*in.  A, 
to  infinity,  in  the  direction  of  the  positive 
abscissas  ;  and  since  these  values  of  y  are 
equal  with  contrary  signs,  it  follows  that 
for  each  assumed  abscissa,  as  AP,  there 
will  be  two  corresponding  points  of  the 
curve,  one  above  and  the  other  below  the 
axis  of  X,  at  equal  distances,  and  the  two  values  of  y  taken  to- 
gether will  form  a  chord,  as  MM',  which  will  be  bisected  by  the 
axis  of  X  ;  hence,  the  curve  is  symmetrical  with  regard  to  the  axis 
ofX. 

The  line  AX  is  called  the  axis  of  the  parabola,  and  the  point  A, 

in  which  it  intersects  the  curve,  is  called  the  vertex  ;  and,   in 

general,  any  straight  line,  which  bisects  a  system  of  chords  perpeU' 

dicular  to  it,  is  an  axis  of  the  curve  in  which  the  chords  are  di*awn. 

If    X  =  0,     we  have 

y  =    d=  0, 

which  proves  that  the  curve  is  tangent  to  the  axis  of  Y,  at  the 
origin,  Art.  (34). 


A 


INDETERMINATE    GEOMETRY". 


105 


If  X  is  negative,  the  values  of  y  are  imaginary  ;  lionce,  there  ia 
no  point  of  the  curve  on  the  left  of  the  axis  of  Y, 
If    y  =  0,     we  have 

a;  =   0, 

and  the  curve  cuts  the  axis  of  X  in  one  point  only,  at  the  origin. 


86.  The  curve  may  be  constructed  by  points  from  its  equation 
as  in  Art.  (22).  This  is  done  geometrically,  thus  :  Let  AX  and 
AY  be  two  co-ordinate  axes  at  right 
angles.  Lay  off  from  the  origin  in 
the  direction  of  the  negative  ab- 
scissas AP'  =  2^,  and  take  any 
positive  abscissa,  as  AP ;  on  the 
line  PP'  as  a  diameter,  describe  a 
circle,  and  from  the  points  in  which 
it  intersects  the  axis  of  Y,  draw 
lines  parallel  to  the  axis  of  X  until  they  intersect  the  perpendicular 
erected  to  AX,  at  P.  The  points  of  intersection,  M  and  M',  will 
be  points  of  the  curve.  For,  from  a  known  property  of  the  circle, 
we  have 


JT^ 

/ 
/ 

^, 

\ 
\ 

X 

Jlf'"^ 

AD*  =  AF   X   AP 


Pi\r 


or 


yg  =  2j?a?.^ 


^' 


8V.     If  a  point  whose  co-ordinates  are  x  and  y,  is  on  the  curve, 
we  must  have  the  condition.  Art.  (23), 


2]px, 


or 


|2     — 


^px  —   0. 


If  the  point  is  without  the  curve,  since  its  ordinate  will  be 
greater  than  the  corresponding  ordinate  of  the  curve,  we  must  have 


y«  >   2par, 


or 


2px  >   0. 


106  INDETERMINATE    GEOMETRY. 

If  the  point  is  within  the  curve, 

y2  <  2px,         or         y«  -   2pa;  <  0. 


88.     If  in  equation   (1),  Art.  (84),  we  make    a;  =  ^ ,    wo 
have 


y'  =  v\ 


T^ 


2y  =   2p. 


Hence,  if  a  point,  as  F,  be  taken  on  the  axis  of  the  parabola,  at 
a  distance  from   the   vertex  equal   to   one 
-:^>'  fourth  of  the  parameter,  the  double  ordinate, 

or  the  chord,  perpendicular  to  the  axis  at  thit 
point,  will  he  equal  to   the  parameter  of  tht 
■^ — ^     curve. 

If  F  be  the  point  and  M  any  point  of 
the  curve,  the  right  angled  triangle  FPM 
will  give 


FM  =   "Vfp'  +  PM', 


or,  since 


FP  =  AP  -  AF  =  a;  _ 


PM  =  y, 


we  have 


FM 


=  \/(—  I- 


+  y^ 


Of  squaring     x  —  IL,    and  substituting  for  y^  its  value  2/>ar, 


FM 


-V 


z*  ^  px  +  SL  =  X  +   ^ 


INDETERMINATE    GEOMETRY. 


107 


If  from  the  vertex  A,  we  lay  off    AB 
perpendicular  to  tlie  axis,  we  sliall  have 


^-,  and  draw  BG 

2 


MC  =  BP 


BA  +  AP  =  ar  4-  ^  =  FM. 

2 


Hence,  the  distance  from  any  point  of  the  curve  to  the  line  BC, 
is  equal  to  the  distance  from  the  same  point  to  the  point  F. 

This  remarkable  property  enables  us  to  define  a  parabola  to  be 
a  curve,  such,  that  each  of  its  points  is  at  the  same  distance  from  a 
given  point  and  a  given  straight  line. 

The  given  point,  F,  is  called  the  focus,  the  given  line  BC,  the 
directrix,  and  a  straight  line  drawn  through  the  focus  perpendicu- 
lar to  the  directrix,  is  the  axis  of  the  parabola. 

This  property,  also,  gives  another  simple  method  of  constructing 
the  curve  by  points,  when  the  directrix 
and  focus  are  given.  Let  BC  be  the  di- 
rectrix and  F  the  focus.  Through  F 
draw  FB  perpendicular  to  BC,  it  will  be 
ttie  axis.  At  any  point  of  the  axis,  as 
P,  erect  a  perpendicular ;  with  the  focus 
F  as  a  centre,  and  radius  BP,  describe 
arcs  cutting  the  perpendicular  in  M  and 
M' ;  these  will  be  points  of  the  curve,  since 


^ 

jy 

^ 

.£' 

[' 

^^ 

i^ 

FM 


BP  =  MC. 


The  curve  may  also  be  constructed  by  a  continuous  movement. 
Place  one  side  DC,  of  a  right  angled  tri- 
angular rule  DCE,  against  the  directrix ; 
fasten  one  end  of  a  string  equal  in  length 
to  the  other  side  EC,  at  the  point  E,  and 
the  other  end  at  the  focus  ;  press  a  pencil 
against  the  string  and  rule,  and  as  the  rule 
is  moved  along  the  directrix,  the  point  of 
the  pencil  will  describe  the  parabola ;  for 
we  always  have 


108 


INDETERMINATE    GEOMETRY. 


FM  =  MC. 


^ 


89.  Let  x\  y'  and  a;",  y"  be  the  co-ordinates  of  any  two  point*- 
of  the  parabola.  Since  these  are  points  of  the  curve,  their  co-ordi- 
nates will  satisfy  its  equation  and  give  the  two  conditions,  Art.  (23), 


y 


Ipx', 


=   2px' 


from  which,  omitting  the  common  multipHer  2p,  we  obtain  the 
proportion 

y'2    :    y"2    ::   x'    :   x'\ 

that  is,  the  squares  of  the  ordinates  of  any  two  points  of  the  curve 
are  proportional  to  the  corresponding  abscissas. 


90.  Let  ic",  "y"  be  the  co-ordinates  of  any  point,  as  M,  on  the 
curve,  and  through  this  point  conceive  any  straight  line  to  be' 
drawn  ;  its  equation  will  be  of  the  form,  Art.  (29), 


y   -  y"  =  d{x  —  x"). 


.(1), 


in  which  d  is  undetermined.     Since  the  given  point  is   on  the 

curve,  we  must  have  the  condition 


y"2    =    2px". 


we  have 


or 


/«  — 


(y  +  y"){y 


Subtracting  this,  member  by  mem- 
ber, from  the  equation 

y*  =   2px, 


=   2p{x  —  x"\ 
y")   =   2p{x  -  x"\ 


which  ig  the  equation  of  the  parabola,  with   the  condition  mtro- 
duced  that  the  given  point  shall  be  on  the  curve.    Combining  this 


INDETERMINATE    GEOMETRY.  109 

with  equation  (1),  by  substituting  the  value  of  y  —  y",  taken 
from  (1),  we  obtain 

(y  +  y")  d{x  -   X")  =   2p{x  -  x'% 

or 

t(y  +  y")d  -   2p]{x  -  X")   =   0 (2), 

m  which  x  and  y  must  represent  all  the  points  common  to  the 
right  line  and  curve,  Art.  (27).  This  equation  being  of  the^second 
degree,  there  are  two  such  j)oints,  and  only  two ;  and  the  equa- 
tion may  be  satisfied  by  placing  the  factors  separately  equal  to 
0.     Placing 

X  —  x"   —   0,  we  have  x  =  x"^ 

and  this  value  in  (1)  gives  y  =  y".  The  values  thus  obtained 
are  the  co-ordinates  of  the  given  point,  which  is  one  of  the  points 
common  to  the  two  lines.  By  placing  the  other  factor  equal  to  0, 
we  have 

(y  -I-  y")d  ~  2\  =  0... (3), 

in  which  y  must  be  the  ordinate  of  the  second  point  of  intersection, 
M'.  If  now,  the  right  line  be  revolved  about  the  point  M,  so  as  to 
cause  the  point  M'  to  approach  M,  y  in  equation  (3),  becomes 
nearer  and  nearer  equal  to  y",  and  finally,  when  the  two  points  co- 
incide, we  shall  have  y  =  y",  the  line  will  be  tangent  to  the 
curve,  and  equation  (3)  reduce  to 

2y"d  =   2jL>,  whence  d  =z  ±- , 

y" 

which  is  the  value  d  must  have  when  the  assumed  line  becomes  a 
tangent.     Substituting  this  value  of  d  in  (1),  we  have 

y  -  /'  =  L.(x  -  x'% 
y" 
or 


110  INDETERMINAIE    GEOMETRY. 

yy"  —  y"^  —  px  -  px", 

which  by  the  substitution  of  2^;^"  for  y"^.  becomes 

yy"  =  2A^  +  ^") (4), 

for  the  equation  of  a  tangent  line  to  the  parabola  at  a  given  point. 


91.     If  we  multiply  both  members  of  the  last  equation  by  2, 
and  subtract  the  result,  member  by  member,  from  the  equation 

y"2   =    2px", 


we  have 


/"«  — 


2yy"  = 


2px, 


H^' 


adding  y^  to  both  members, 

y"«   -    2yy"   -{-  y»  =  y^  —  2px, 
or 

{y"  —  yY  =  y^  -  2px, 

The  first  member  being  a  perfect  square,   is  positive  for    all 
values  of  y  except    y  =  y"  ; 

y*  —   2px, 

is  therefore  positive  for  all  values  of  y  and  x^  except  y  =  y", 
X  —  x"y.  when  it  will  be  0  ;  hence,  since  x  and  y  are  the  gener- 
al co-ordinates  of  the  tangent,  all  points  of  the  tangent^  except  the 
point  of  contact,  are  without  the  curve,  Art.  (87). 

92.     If  in  equation  (4),  Art.  (90),  we  make     y  =  0,  we  find 

A. 

y\        2^^^^^^^^^         0  =  j9  (a;  -f  x'%     or     a;  =  —  a:", 

V    for  the  distance  AT,  to  the  point  in 
which  the-  tan 
hence,  we  have 

Yl  =  TA  +   AP 


^ 


which  the-  tangent    cuts  the  axis ; 


INDETERMINATE    GEOMETRY.  Ill 

The  distance  PT  is  called  the  suhtangent,  which,  in  general,  is 
the  distance  from,  the  foot  of  the  ordinate  of  the  point  of  contact^  to 
the  point  in  which  the  tangent  cuts  the  axis,  to  which  the  ordinate 
is  drawn  ;  and  in  the  parabola,  is  equal  to  double  the  abscissa  of 
the  point  of  contact. 

This  property  gives  a  simple  method  of  drawing  a  tangent  to  a 
|»arabola  at  a  given  point.  Let  M  be  the  point.  From  the  vertex 
lay  off,  on  the  axis  Avithout  the  parabola,  a  distance  AT,  equal  to 
the  abscissa  of  the  given  point ;  draw  a  right  line  from  the  ex- 
tremity of  this  distance  to  the  point  of  contact,  it  will  be  the  re- 
quired tangent.  ^ 

93.  If  the  point  M  be  joined  with  the  focus  F,  we  have, 
Art.  (88), 

FM   =  X"   +  I.  rf 


But  since   AT 
we  also  have 


=  x^\    and  AF=-^,     -i^.   t  f — L 
'  2  5'    -p-^V^       \ 


Yl  =  x"   +  ^ ; 


o 


hence,    FM  =  FT,     the  triangle  TFM  is  isosceles,  and  the  angle 
FMT  =  FTM. 

Hence,  if  a  right  line  be  drawn  from  the  focus  of  a  parabola  to 
the  point  of  contact  of  a  tangent,  this  line  will  maJce  an  angle  with 
the  tangent  equal  to  that  tohich  the  tangent  makes  with  the  axis. 

This  property  enables  us  to  make  the  following  constructions. 

First.  To  draw  a  tangent  to  the  parabola  at  a  given  point. 
Draw  a  right  hne  from  the  point,  as  M,  to  the  focus ;  with  this 
line  as  a  radius  and  the  focus  as  a  centre,  describe  an  arc  cutting 


112  INDETERMINATE    GEOMETRY. 

•  • 

the  axis,  without  the  curve,  in  a  point,  as  T ;    draw  a  right  line 

from  this  to  the  given  point,  it  will  be  the  required  tangent,  as  the 

triangle  MFT  will  be  isosceles. 

Or  otherwise,  thus.     Draw  a  right  line  through  the  given  point 

perpendicular  to  the  directrix  ;  join  the  point  C,  in  which  it  inter- 
sects the  directrix,  with  the  focus, 
and  through  the  given  point  draw 
a  right  line  perpendicular  to  this 
last  line,  it  will  be  the  tangent. 
For,  since  MF  =  MC,  the  tri- 
angle CMF  is  isosceles  and  there- 
fore the  angle  FMT  =  CMT ; 
GMT  =  MTF;     hence, 

FMT  =  MTF. 

Second.  To  draw  a  tangent  from  a  point  without  the  curve,  as 
N.  Join  the  point  with  the  focus  ;  with  this  distance  as  a  radius, 
and  the  given  point  as  a  centre,  describe  an  arc  cutting  the  direc- 
trix in  the  points  C  and  0' ;  through  these  points,  draw  lines  par- 
allel to  the  axis,  cutting  the  curve  in  the  points  M  and  M' ;  join 
these  points  with  the  given  point  and  we  shall  have  the  tangents 
NM  and  NM'.     For,  sine* 

MF  =  MC,  and  NF  =  NC, 

the  line  NM  has  two  of  its  points  equally  distant  from  the  points 
F  and  C,  is  therefore  perpendicular  to  FC  at  its  middle  point  and 
bisects  the  angle  FMC. 

Let  the  co-ordinates  of  the  given  point  N,  be  denoted  by  x'  and 
y'.  Since  this  point  is  on  the  tangent,  we  must  have  the  equation 
of  condition.  Art.  (23), 

y'y"  =  p{x'  -f  X") (1), 

and  since  the  point  of  contact  is  on  the  parabola,  we  also  have  the 
equation  of  condition, 

y"^  =   2px". 


INDETERMINATE    GEOMETRY.  113 

« 

In  these  equations  x"  and  y"  are  unknown,  and  since  .one  is  of 
the  first  and  the  other  of  the  second  degree,  their  combination 
will  give  an  equation  of  the  second  degree,  and  there  will  be  two 
values  of  x"  and  two  corresponding  of  y". 

Combining  these  equations  by  substituting  the  value 

x"    =    1—, 

2p 
m  the  first,  we  obtain 

y/2   _    2y'y"   =    -   2px' (2), 

from  which  we  deduce  the  two  values 


7j"  =  y'   dcz    vi/2~ir~22^ 

The  values  of  y"  will  evidently  be  real,  when 

y'^  —   2px'  >   0, 

that  is,  when  the  given  point  is  without  the  curve,  Ai-t.  (87),  and 
there  will  be  two  tangents,  as  appears  by  the  geometrical  construction. 

The  values  of  y"  will  be  equal  when  the  point  is  on  the  curve 
and  there  will  be  but  one  tangent. 

They  will  be  imaginary  when  the  point  is  within  the  curve  and 
there  will  be  no  tangent.  " 

The  corresponding  values  of  x"  being  found,  each  set  of  co-or- 
dinates may  be  substituted,  in  succession,  in  equation  (4),  Art.  (90), 
and  the  equations  of  the  two  tangents  thus  determined. 

Third.  To  draw  a  tangent  parallel  to  a  given  line  as  SR.  Pro- 
duce the  line  until  it  intersects  the  axis  at  S,  with  the  focus  as  a 
centre,  and  the  distance  FS  as  a  ra- 
dius describe  an  arc  cutting  the 
given  line  in  R,  join  this  point  with 
the  focus,  the  point  M,  in  which  the 
last  line  intersects  the  curve  will  be 
the  point  of  contact,  through  which 
draw  MT  parallel  to  the  given  line, 
8 


114 


INUKTKRMINATE    GEOMETRY. 


it  will  be  the  required  tangent.     For,  since  MT  is  parallel  to  RS, 
and     FS  =  FR,     we  have 

FM  =  FT. 


94.  Since  the  triangle  FMT  is  isosceles,  the  line  FD,  drawn  per- 
pendicular to  the  base  MT,  will  pass 
through  its  middle  point;  and  since 
AT  =  AP,  Art.  (92),  the  line  AD 
also  passes  through  the  middle  point 
of  MT  :  Ilence,  if  from  the  focus  of  a 
2mrahola,  a  right  line  be  drawn  perpen- 
dicular to  any  tangent^  it  icill  intersect  this  tangent,  on  the  tangent 
at  the  vertex  ;  and  conversely. 

Since  the  triangle  FDT  is  right  angled  at  D,  we  have 

FD*    ==  AF  X  FT,  ^ 

and  since  AF  is  constant  and  FT  =  FM  ;  the  square  •  of  the 
perpendicular  FD,  will  vary  as  the  first  power  of  the  distance  from 
the  focus  to  the  point  of  contact. 


95.     If  in  equation  (1),  Art.  (93), 

y'y"   =z  p  [x'   +    x"). 


.(1), 


we  regard  x"  and  y''  as  variables,  it  will  be  the  equation  of  a  right 

line,  Art.  (25) ;  and  since  both 
values  of  x"  and  y"  deduced 
from  equation  (2),  Art.  (93), 
must  satisfy  this  equation,  the 
line  represented  by  it  will  pass 
through  both  points  of  contact, 
and  will  therefore  be  the  inde- 
finite  chord  which  joins  these 


INDETERMINATE    GEOMETRY'.  115 

points.  If  any  point  as  O,  be  taken  upon  this  chord,  its  co-ordinates, 
which  we  will  denote  by  c  and  c7,  when  substituted  for  x"  and  y" 
will  satisfy  the  equation  and  give  the  condition 

y'd  =  p(x'   +  c) (2). 

Now  it  is  evident,  that  every  set  of  values  for  x'  and  y'  which 
will  satisfy  this  equation,  will  give  a  point  from  which,  if  two 
tangents  be  drawn  to  the  parabola  and  the  points  of  contact  be 
joined  by  a  chord,  this  chord  will  pass  through  the  point  O. 
Hence,  if  y'  and  x'  be  regarded  as  variables  in  this  equation,  it 
will  represent  a  right  Hne  every  point  of  which  will  fulfil  the  above 
condition. 

This  line  is  called  the  polar  line  of  the  point  0,  which  is  called 
the  pole. 

If  through  the  point  O,  a  line  be  drawn  parallel  to  the  axis  AX, 
the  ordinate  of  the  point  in  which  it  intersects  the  curve  will  be 
equal  to  d,  and  the  equation  of  a  tangent  to  the  parabola,  at  this 

point,  will  be.  Art.  (90), 

« 

yd  —  p{x  +  x"), 

and  this  tangent  is  evidently  parallel  to  the  line  represented  by 
equation  (2),  that  is  to  the  polar  hne.  Art.  (28), 

If  the  line  OA'  be  further  produced  till  it  intersects  the  polar 
line  in  N,  the  ordinate  of  this  point  will  be  d,  which  substituted 
for  y'  in  equation  (1),  will  give 

y'>d  =  p  {x'  4-  x"), 

for  the  equation  of  the  chord  corresponding  to  this  poiiXt  N,  and 
this  is  parallel  both  to  the  polar  line  and  tangent. 
Tliese  properties  give  the  following  constructions  : 
1.  The  pole  being  given,  to  construct  the  corresponding  polar 
line. 

Through  the  pole  draw  a  line  parallel  to  the  axis  of  the  para- 
bola ;  at  the  point  in  which  this  intersects  the  curve,  draw  a 


116  INDETERMINATE    GEOMETRY, 

tangent ;  through  the  pole  draw  a  chord  parallel  to  this  tangent, 
and  at  its  extremity  draw  another  tangent ;  through  the  point  in 
which  this  intersects  the  line  first  drawn,  draw  a  line  parallel  to  thft 
chord,  it  will  be  the  polar  line. 

2.  The  polar  line  being  given,  to  construct  the  pole. 

Draw  a  tangent  parallel  to  the  polar  line  ;  through  the  point 
of  contact  draw  a  line  parallel  to  the  axis  ;  from  the  point  in  which 
this  intersects  the  polar  line,  draw  another  tangent ;  through  the 
point  of  contact  thus  determined,  draw  a  chord  parallel  to  the 
polar  line,  it  will  intersect  the  line  parallel  to  the  axis  in  the  re- 
quired pole. 


'/S'     96.     If  the  focus  be  taken  as  the  pole,  the  co-ordinates  of  which 
are 

d  =   0,  c  z=  ^. 

2 

equation  (2)  of  the  preceding  article  reduces  to 

0=  p{x'  +  -|-),  or         X'  =  ^  ^, 

y'  heing  indeterminate^  which  is  the  equation  of  the  directrix,  Art. 
(21).  The  directrix  is  then  the  polar  line  of  the  focus.  Hence,  if- 
any  chord  he  drawn  through  the  focus  of  a  parabola  and  two  tan- 
gents he  drawn  at  its'  extremities ^  these  tangents  will  intersect  on  the 
directrix.  -     A 

97.  If  in  the  general  equation  of  a  right  line  passing  through 
a  given  point.  Art.  (29),  we  substitute  for  x'  and  y',  the  co-ordi- 
nates of  the  focus,  we  shall  have 

>  y  =  a{x-L.) (1), 


INDETERMINATE    GEOMETRr.  117 

for  the  equation  of  any  chord  passing  through  the  focus.     Com- 
bining this  with  the  equation  of  the  parabola,     y"^  =   2px^     by 

substituting  the  vahie    a;  =  jL  ,    we  have 

2p 

a{yl-l.\  or  y^-^Pl^pK 

Denoting  the  two  roots  of  this  equation  by  y'  and  y",  we  have 
from  a  well  known  principle  of  Ali^ebra,,  «/       "  /  r- 

y'y"  =  -  i^\ 

and  if  d  and  d'  denote  the  tangents  of  the  angles  made   with  the 
-jZ  axis,  by  two  tangents  drawn  at  the  extremities  of  this  chord,  we 


have,  Art  (90), 


whence. 


d=P.,  d'-^l.', 

y'  y" 


dd'  =  ^, 


or  substituting  for  y'y"  the  above  value, 

dd'  —   —   1,         or         dd'  '\-   1   =z  0. 

Hence,  Art.  (28),  if  at  the  extremities  of  a  chord  passing  through 
the  focus  of  a  parabola^  tioo  tangents  he  drawn^  they  will  he  perpen- 
dicular to  each  other,  and  intersect  on  the  directrix,  Art.  (96). 

And  conversely,  if  ttoo  tangents  to  the  parahola  are  perpendicular 
to  each  other,  the  chord  joining  their  points  of  contact  wiU  pass 
through  the  focus.  For,  let  S'M  and 
S'M"  be  the  two  tangents.  If  the 
chord  MM"  does  not  pass  through  the 
focus ;  through  the  focus  and  the  point 
M,  draw  MM' ;  at  M'  draw  the  tangent 
M'S.  From  what  precedes,  it  must 
be  perpendicular  to  MS' ;  hence,  SM' 


118  INDETERMINATE    GEOMETRY, 

and  S'M''  must  be  parallel.  But  since  the  tangent  of  the  angia 
which  a  tangent  to  the  parabola  makes  with  the  axis  is    J^ ,  Art. 

y" 

(90),  no  two  tangents  can  be  parallel,  for  no  two  points  have  equal 
ordinates.  It  is  then  absurd  to 'suppose  that  MM''  does  not  pass 
through  F. 

98.     Through  the  point  of  contact  of  a  tangent,  let  anj  other 
straight  line  be  drawn,  its  equation  will  be  of  the  form.  Art.  (29), 

y  -   y"   =  d'{x  -  X") (1). 

If  this  line  is  perpendicular  to  the  tangent,  we  must  have,  Art 
(28), 


dd'   + 

1 

=   0, 

or 

d' 

= 

But, 
lence 

Art.  (90), 

d 
d> 

y" 

Substituting  this  in  equation  (1),  we  have 

y  -  y"  =   -  '^{x  -  X") (2), 

P 

for  the  equation  of  a  straight  line  perpendicular  to  the  tangent  at 
the  point  of  contact.     This  line  is  called  a  normal. 
If  we  make    y  =  0,     in  equation  (2),  we  have 

-  y"  =  -  yl{x  _  x"), 
P 

or 

X         —         X"         =■       Py 


INDETERMINATE    GEOMETRY. 


119 


ill  whicli,  X  is  the  distance  AR  from  the  origin  to  the  point  in 
which  the  normal  intersects  the  axis,  and 

a:  —  a;"  =  AR  -  AP  =  PR  =  IX 

The  distance  PR,  from  the  foot  of  the 
ordinate  of  the  point  of  contact,  to  the 
point  in  which  the  normal  cuts  the  axis,  is 
called  the  subnormal.  Hence,  the  subnor- 
mal in  the  parabola  is  constant  and  equal  to  half  the  parameter  of 
the  curve. 

This  property  enables  us  to  construct  a  tangent  at  a  given  point. 

Draw  the  ordinate  of  the  point ;  from  its  foot  lay  off  a  distance 
equal  to  one  half  the  parameter  ;  join  the  extremity  of  this  dis- 
tance with  the  given  point,  through  which  draw  a  perpendicular 
to  the  last  line,  it  will  be  the  required  tangent. 


OF   THE    PARABOLA    REFERRED    TO    OBLiaUE  CO-ORDINATE 

AXES. 


99.  It  was  observed  in  Art.  (71),  that  two  classes  of  proposi- 
tions might  arise  in  the  transformation  of  co-ordinates.  As  an  ex- 
ample of  the  second  class,  let  it  now  be  proposed  to  ascertain  if 
there  are  any  other  co-ordinate  axes,  to  which  if  the  parabola  be 
referred,  its  equation  will  be  of  the  same  form  as  when  referred,  to 
its  axis  and  the  tangent  at  its  vertex. 

For  this  purpose,  let  us  take  the  general  formulas  (3),  Art.  (67), 

X  =  a  -{-  x'  cos  a   -^  y'  cos  a', 
y  =  6   -{-  a:'  sin  a   -|-  y'  sin  a', 
and  substitute  the  values  of  x  and  y  in  the  ec  nation 


r 


'Zpx. 


.(X). 


We  thus  obtain 


120  INDETERMINATE    GEOMETRY. 

b^  +  2bx'  sin  a  -f-  x'^  sin*  a  -{-  26?/'  sin  a'  +  2a;'y'  sin  a  sin  a,' 

+ J^!?-Sin^  a'  =  2pa  +   2/>x''  cos  a   +   2p7/'  cos  a', 

or  transposing,  arranging  and  omitting  tlie  dashes  of  tlie  variables, 

2/2  sin*  a'   +  ^*  sin*  a   +   2x](  sin  a  sin  a' 

+  2(6  sin  a'  —  p  cos  a')?/  +  6*  —  2pa  =  2[pcoa  a  —6  sin  a)x...(2), 

which  is  the  equation  of  the  parabola  referred  to  any  oblique  axes. 
In  order  that  this  equation  shall  be  of  the  same  form  as  equation 
(1),  the  absolute  term,  in  the  first  member,  and  the  terms  contain- 
ing a;*,  xy,  and  y,  must  disappear,  ^Yhich  requires  that 

6*  —   2pa  =  0 (3); 

sin*  a  =  0 (4); 

sin  a  sin  a'  =  0 (5)  ; 

6  sin  a'  —  ^  cos  a'  =  0 (6). 

These  equations  contain  four  arbitrary  constants,  it  is  therefore 
possible  to  assign  such  values  to  the  constants  as  to  satisfy  the  four 
equations,  and  thus  reduce  equation  (2)  to  the  proposed  form. 

Equation  (3)  is  the  equation  of  condition  that  the  new  origin 
shall  be  on  the  curve,  Art.  (87). 

Equation  (4)  can  only  be  satisfied  by  a  =  0,  or  =1 80°  ; 
hence  the  new  axis  of  X  must  he  parallel  to  the  axis  of  the  curve. 

Equation  (5)  is  satisfied  by  sin  a  =  0,  without  introducing 
any  new  condition. 

Equation  (6)  can  be  put  under  the  form 

sin  cl'         .  f         p 

=  tans:  a'         ^ 


cos  a 


b 


and  therefore,  Art.  (90),  expresses  the  condition  that  the  new  axis 
^ofY  shall  be  tangent  to  the  curve. 


INDETERMINATE    GEOMETRY.  121 

Since  we  have  thus  far  introduced  but  three  independent  con- 
ditions, and  since  a,  b  and  a'  are  still  undetermined,  we  may  assign 
a  value  at  pleasure  to  either  of  them,  whence  the  other  two  will 
become  known,  and  an  injiniie  number  of  sets  of  co-ordinate  axes 
be  thus  determined,  which  will  fulfil  the  required  condition, 
each  of  which  will  be  subjected  to  the  three  conditions  expressed 
by  equations  (3),  (4)  and  (6). 

Substituting  the  above  conditions  in  equation  (2),  and  observing 
that,  since     sin  a  =   0,     cos  a  =   1,     we  have 

N.    2/2  sin^  cc'  =   2px,         or  y^  =  — ^  x  ; 

/-^  sin^  a' 

or,  denoting  the  coefficient  of  x  by  2p' 

y^  =   2p'^ (V), 

an  equation  of  the  same  form  as  (1). 


100.     Solving  the  last  equation  with  reference  to  y,  we  find 

y  =   =fc   V2p% 

and  we  see,  as  in  Art.  (85),  that  every  positive  value  of  x,  gives 
two  real  values  of  ?/,  equal  with  contrary 
signs,  and  that  these  two  values  taken  to- 
gether form  a  chord,  as  MM',  parallel  to 
the  axis  of  Y,  which  chord  is  bisected  by 
the  axis  of  X,  at  P.  The  hne  A'X  is 
therefore  called  a  diameter  of  the  parabola; 
and,  in  general,  any  straight  line  ivliich  bi- 
sects a  system  of  parallel  chords  is  a  diameter,  of  the  curve  in  which 
the  chords  are  drawn.  The  points  in  which  a  diameter  intersects 
the  curve  are  called  the  vertices  of  the  diameter. 

Since  condition  (4)  of  the  preceding  article  requires  the  new 
axis  of  X  to  be  parallel  to  the  axis  of  the  curve,  it  follows  that  all 
the  diameters  of  the  parabola  are  parallel  to  each  other. 


122 


INDETERMINATE    GEOMETRT. 


Since  condition  (G)  requires  the  new  axis  of  Y  to  be  tangent  to 
the  curve  at  the  origin,  it  also  follows  that  each  diameter  bisects  a 
system  of  chords  parallel  to  the  tangent  at  its  vertex. 

If  the  parabola  is  given,  traced  upon  paper,  a  diameter  may  be 
found  by  drawing  any  two  parallel  chords  as  MM'  and  bisecting 
them  by  a  straight  line  as  PP  ;  this  line  will  be  a  diameter. 

Draw  any  two  chords  perpendicular  to  this  diameter  and  bisect 
them  by  a  straight  line,  this  will  be 
the  axis.  Art.  (85).  At  the  vertex 
of  the  first  diameter,  A',  draw  a  line 
parallel  to  the  chords  which  it  bi- 
sects, it  will  be  a  tangent  to  the 
curve.  At  the  vertex.  A,  of  the 
parabola,  draw  a  line  perpendicular 
to  the  axis,  it  will  also  be  a  tangent. 
At  the  point  D,  where  these  tangents  intersect,  draw  a  perpendi- 
cular to  the  first,  it  will  intersect  the  axis  in  the  focus  F,  Art.  (94). 
The  property,  that  each  diameter  bisects  a  system  of  chords 
parallel  to  a  tangent  at  its  vertex,  suggests  the  following  construc- 
tion for  drawing  a  tangent  parallel  to  a  given  line,  as  BC.  Draw 
two  chords  parallel  to  the  given  line ;  bisect  them  by  a  straight 
line  PP,  and  at  the  point  A',  where  this  intersects  the  curve,  draw 
a  line  parallel  to  the  given  line,  it  will  be  the  required  tangent. 


101.  The  coefiicient  2/  in  equation  (7),  Art.  (99),  is  called  the 
parameter  of  the  diameter  A'X,  and,  as  in  Art.  (84),  is  a  third 
proportional  to  any  ordinate  and  its  corresponding  abscissa. 

If  we  represent  the  distance  FA'  by  r,  and  recollect  that  the 

angle  FA'D  =  FTD  is  denoted 
by  a',  Art.  (99),  we  shall  have  in 
the  right  angled  triangle  FI^A' 

FD  =  rsina'. 


or 


INDETERMINATE    GEOMETRY.  123 

FD*   =  r2siii«a'. 
But  we  also  have,  Art.  (94). 

FD*  =  FA  X  FA',      or  FD'  =  •?  r. 

2 

I       Equating  these  two  values  of  FD*,  we  have 

r*  sin'^  a'  =:  -L  r  ;  whence  sin^  aJ  z=.  —. 

2  2r 

Substituting  this  value  of  sin'*  a',  in  the  expression,  Art.  (99), 

sin*  a' 

it  reduces  to 

2^  =  4r 

that  is,  </ie  ^parameter  of  any  diameter  of  the  parabola,  is  equal  to 
four  times  the  distance  from  the  vertex  of  the  diameter  to  the  focus. 

102.  Let  a;"  and  y"  be  the  co-ordinates  of  any  point  of  the 
parabola  referred  to  the  diameter   A'X  J 

and  the  tangent  A'Y.  The  equation  of 
a  right  Hne  passing  through  this  point 
will  be 

y  —  y"  =.    d{x  —  x'% 

in  which  d  will  represent  the  ratio  of  the 

sines  of  the  angles  which  the  line  makes  with  the  co-ordinate 

axes,  Art.  (20). 

By  a  process  identical  with  that  pursued  in  Art.  (90),  we  can 
find  the  value  of  d,  when  the  line  becomes  a  tangent,  and  thus  de- 
duce the  equation  of  the  .angent, 


124 


INDETERMINATE    GEOMETRY. 


yy"  =  p'{x  +  x"). 

By  making     y  =  0,     we  find 

X  =   —  x"  =  A'T; 

hence  as  in  Art.  (92),  the  subtangent  PT,  is  equal  to  twice  the  ah- 
scissa  of  the  point  of  contact.  And  a  tangent  may  be  drawn  at  a 
given  point  by  drawing  the  ordinate  MP,  of  the  point,  parallel  to 
the  axis  A'Y,  laying  off  the  distance  A'T  equal  to  the  abscissa 
A'P,  and  joining  the  extremity  of  this  distance  with  the  given 
point 


103.     Let  AM  be  an  arc  of  a  parabola,  in  which  inscribe  any 

polygon,  as     AM' MP.     At  the  points  M,  M',  &c.,  draw  the 

tangents  MT,  M'T',  &c. 
Through  the  middle  points 
of  the  chords  MM',  &c., 
draw  the  diameters  RS, 
2'A^p'         jB'        jp  R"S',  &c.,  and  draw  the 

ordinates  MP,  M'P',  &c.  It  is  evident  that  for  each  chord  there 
will  be  a  trapezoid,  as  MM'P'P,  within  the  parabola,  and  a  corres- 
ponding triangle,  as  OTT',  without. 

Since  the  points  of  contact  M  and  M',  when  referred  to  the  di- 
ameter RS  and  tangent  line  at  its  vertex,  have  the  same  ab&cds&a 
3[R,  the  subtangent  will  be  the  same  for  each.  Art.  (102),  and  the 
two  tangents  MO  and  M'O  will  intersect  the  diameter  VS,  at  the 
same  point  0 ;  hence  the  altitude  of  the  triangle  OTT'  will  be 
equal  to  the  line  RR',  drawn  through  the  middle  points  of  the  two 
inclined  sides  of  the  trapezoid  P'M'MP  ;  and  since, 


AP 


AT, 


and 


AP 


AT', 


we  have 


PF 


T£' 


'^■u.  fouqj^  in 


INDETERMINATE    GEOMETRY.  125 

hence,  the  area  of  the  trapezoid,  which  is  measured  by 

RR'   X  PP', 
is  double  the  area  of  the  triangle,  which  is  measured  by 

IrR'    X   TT'; 
2 

and  so  for  each  trapezoid  and  corresponding  triangle,  and  the  sum 
of  all  the  interior  trapezoids  will  be  double  the  sum  of  the  corres- 
ponding triangles. 

If  now,  the  number  of  sides  of  the  polygon  be  increased  indefi- 
nitely, the  sum  of  the  trapezoids  will  be  the  same  as  the  curvilin- 
ear area  AM'MP,  and  the  sum  of  the  triangles  the  same  as  the 
exterior  area  TMM'A  ;  hence  the  first  area  is  double  the  second. 
But  the  sum  of  these  ]bwo  areas  is  equal  to  the  area  of  the  triangle 
MTP,  therefore 

AM'MP  =  hlTF. 
3 

But  since     TP  =   2AP,     we  have 

triangle  MTP  =  rectangle  ALMP. 

Therefore,  the  area  of  a  portion  of  the  parabola  is  equal  to  Uco 
thirds  of  the  rectangle  described  on  the  ordinate  and  abscissa  of  the 
extreme  point. 


^ 


OF    THE    POLAR    EaUATION    OF   THE    PARABOLA. 

104.     Let  us  resume  the  equation 
2/2  =   2px, 

and  substitute  for  y  and  x,  their  values  taken  from  the  formulai 
(2)  of  Art.  (69)  ; 

a;  =  a'   -f-  r  cos  y,  y  =  6'  -}-  r  sin  v. 


126  INDETERMINATE    GEOMETRY. 

We  thus  obtain 

h'^  +   2  h'r  sin  v  •\-  r^  sin'  v  —   2p[a'  -\-  r  cos  v), 
or  transposing  and  arranging, 
r*  sin«  V   +   2  (6' sin  V  —  ^  cos  v)  r  +  h'^  —  2pa'  =   0 (1), 

whicli  is  the  general  2>olar  equation  of  the  parabola^  Art.  (69). 

By  assigning  particular  values  to  a'  and  6',  the  pole  may  be 
placed  at  any  point  in  the  plane  of  the  curve. 

First.  If  it  be  required  that  the  pole  shall  be  on  the  curve,  we 
must  have.  Art.  (87), 

6'2  —   2pa'   =  0, 

and  equation  (1)  reduces  to 

\r  sin*  V  +   2(6'  sin  v  —  p  cos  v)\r  =   0, 

which  may  be  satisfied  by  placing     r  =  0,     or 

r  sin*  V  +  2(6'  sin  v  —  p  cos  v)  =  0 (2). 

Since  the  pole  is  on  the  curve,  as  at  P,  it  is  evident,  that  one 
value  of  ?*_^hould.  be  0,  whatever  be  the 
value  of  V  ;  and  that  the  other  value,  de- 
duced from  equation  (2),  should,  as  v  is 
changed,  give  the  distance  of  each  point  of 
the  curve  from  the  pole  P. 

If  the  point  M  is  moved  along  the  curve 
until  it  coincides  with  P,  the  second  value 
of  r  will  become  0,  and  equation  (2)  will  reduce  to 

b'  sin  V  —  p  cos  v  =   0, 


or 


sm  V 
cos  V 


tanjr  V  z= 


INDETERMINATE    GEOMETKV.  127 

as  it  should,  Art.  (90),  since  the  radius   vector  will  now  coincide, 
in  direction,  with  the  tangent  PT. 

Second.     If  the  pole  be  placed  at  the  focus,  we  must  have 


a'  =  Z.,  6'  =  0, 


and  these  values,  in  equation  (1),  give 

r'  sin*  V  —   2p  cos  vr  —  p*  =.   0, 
and  after  transposing  jp*  and  dividing  by  sin'  v, 


2p  cos  V    ,  _ 


sin^  V  sin'  v 

Solving  this  equation,  we  have 


or 


p  cos  "0    y^  A  I   V^            V^  cos*  V 
—   —r-^ ±   V  -T-^ —    H —^ » 


p  cos  "0         .  A>*  (sin''  V  ■\-  cos^  v)  p  cos  v  dr  jp 

sin*  V  ^  sin^  v  sin*  v 

since     sin*  v  +  cos*  v  =   1. 

As  the  cos  V  inu<5t  be  less  than  radius  or  unity,  we  have 

/)  cos  V  <  p^ 

and  the  second  value  of  r  is  always  negative^  and  must  therefore, 
be  rejected.  Art.  (09).  The  first  value  may  be  placed  under  the 
form 

p  (cos  V  +   1) 

r       =       ;_ , 

sm'  V 
and  since 


128  INDETERMINATE    GEOMETRY. 

sin*  V  =  1  —  COS*  V  =  [l   -{■  COS  v)  (l   —  cos  v), 
it  may  be  still  furtlier  reduced  to 


•(3), 


r  = P 

1   —    COS  V 


which  is  positive  for  all  values  of  v. 

If    V  =  0,     cos  V  =   Ij     and  the  value  of  r  becomes 


0 


and  the  radius  vector  takes  the  direction  AX,  and  gives  that  2)oint 
of  the  curve  which  is  at  an  infinite  distance. 

li    V  =   90°,    cos  V  =  0,    and  the  value 
of  7'  becomes 


~^.^ 


r  =  p  =  FM. 
If    V  =   180°,     cos  v  =   —   1,     and 


=  Z.  =  FA. 
2 


Thus  by  varying  v  from  0  to  360°,  all  the  points  of  the  para- 
bola may  be  determined. 

If  we  wish  to  estimate  the  variable  angle  from  the  Hne  FA,  to 
the  right,  instead  of  in  the  usual  way,  from  the  hne  FX  to  the 
left,  we  have  simply  to  change  v  into  180°  —  -y',  in  which 
case  cos  V  =  —  cos  v',  and  the  value  of  r,  equation  (3),  be- 
comes 


1   +  cos  v' 


in  which    v'  =  0,    gives    r  =   j^,    and     v'  =  180°,    gives 


INDETERMINATE    GEOMETRY.  129 

OF    THE    ELLIPSE    AND    HYPERBOLA. 

105.     We  have  seen,  Art.  (83),  that  the  equation 

y%  _  ,.2^2   ^   2i;a',         or         if  =   1]px  +  r'^jc^ (1), 

represents  the  elHj^se  when  ?•-  is  negative,  and  the  h3'perbola  when 
it  is  positive. 

This  equation  being  of  the  second  degree,  the  ellipse  and  hyper- 
bola are  both  lines  of  the  second  order,  Art.  (33). 

If  in  the  equation,  we  make     x  =  0,     we  find 

y   =    ±   0; 

iience  the  axis  of  Y  is  tangent  to  each  curve,  at  the  origin  of  co^ 
ordinates,  Art.  (34). 

If  we  make     ?/  =   0,     we  have 

2px  +  r2^2  =   0,  or         x{2p  -f  r'^x)  =  0, 

\^hich  may  be  satisfied  by  making 

X  =  0, 
or 


2p  -f  r^x  =  0 ;         whence         x  = 


2;>. 


hence  each  of  the  curves  intersects  the  axis  of  X  in  two  points, 
one  at  the  origin,  and  the  other  at  a  distance  from  it  equal  to 
__   2p 

Now  let  us  transfer  the  origin  of  co-ordinates,  to  a  point  on  the 
axis  of  X,  at  a  distance    —  —  ,     equal  to  half  the  distance  from 

y.2 

the  origin  to  the  second  point  in  which  the  curve  cuts  the  axis ; 
the  new  axes  being  parallel  to  the  primit'  re.     In  formulas  (2),  of 
Art.  (67),  we  must  then  have 
9 


130  INDETERMINATE    GEOMETRY. 


o'    =  _  4,  6-  =  0, 


and  tlie  formulas  become 


^'       y  =  y- 


Substituting  these  values  of  a;  and  y  in  equation  (1),  we  have 


or  reducing  and  omitting  the  dashes, 


yi  ^  ,,2^8  __   fi (2). 


If  in  this  we  make     y  =  0,     we  have 


^^  =  ^! (3),  or  a:  ==   ±  4; 

hence,  each  curve  now  intersects  the  axis  of  X  in  two  points,  one 
on  the  right  and  the  other  on  the  left  of  the  origin,  at  equal  dis- 
tances from  it. 

If    X  =  0,     we  have 


7/a  =   -  il 


(4),         or  y  =   db  \/-  ^, 

V  y.2 


and  these  values  of  y  will  be  ronl  for  the  ellipse,  and  imaginary  for 
the  hyperbola  ;  hence,  the  ellipse  intersects  the  axis  of  Y  in  two 
points,  at  equal  distances  from  the  origin,  one  above  and  the  other 
below  the  axis  of  X  ;  and  the  hyperbola  does  not  intersect  the  axis 
of  Y. 

Giving  to  r*  its  negative  sign  for  the  ellipse,  expressions  (3)  and 
(4)  will  be  essentially  positive,  and  we  may  write 


INDETERMINATE    GEOMETRV.  131 

from  which,  by  deducing  the  values  of  2^^  and  equating  them,  wo 
have 

aV  =.    ~   r26«,  or  r«  =    —   ^; 


and  substituting  this  in  either  of  the  above  equations,  we  find 

b*  b-» 

p^  =  —.  or         p     =—. 

a*  a 

By  the  substitution  of  these  values  of  r*  and  p"*  in  equation  (2), 
and  reducing,  we  have  the  equation  of  the  ellipse, 

ahf  +  6«jc«  ==  «»6* {e). 

For   the    hyperbola     —  —    is  essentially  negative,  we  must 

then  place  it  equal  to  —  6*,  while  the  expression  for  a'  will  remain 
unchanged.  If  then,  in  the  above  equation,  we  simply  change  6^ 
into    —  l)^,    we  obtain  the  eqtiation  of  the  hyperbola, 

a'if  _   6%«  =   _   a%^ (A). 

Furthermore,  it  is  evident  from  the  preceding  discussion,  that 
any  expression  containing  5,  belonging  to  the  ellipse,  will  become 
the  corresponding  one  for  the  hyperbola,  by  changing  6*  int^ 
—   t*,    or  b  into     b  V—    1. 


^  n 


106.     Solving  equation  (e)  with  reference  to  y,  we  have 

"' = 5^"'  -  ^'^'      ^  =  =*=  T  v^"^^^ (1), 


132 


INDETERMINATE    GEOMETRY. 


«  X 


in  which  every  value  of  x  numerically  less  than  a,  whether  positive 

or  negative,  gives  two  real  values  of  y  equal  with  contrary  signs  : 

Ilence,  C  being  the  origin,  CX  and 
CY  the  axes  of  co-ordinates,  and 
CB  and  CA  each  numerically  equal 
to  a,  the  curve  is  continuous  be- 
tween the  points  A  and  B;  and 
since  each  set  of  the  equal  values 
of  2/  forms  a  chord  as  MM',  which 

is  bisected  by  the  axis  of  X,  the  curve  is  symmetrical  with  respect 

to  the  line  AB. 

X  =   +  a     or     —   a,     gives 

2/  =   db  0; 

hence  the  ordinates  at  the  points  A  and  B,  when  produced,  are 
tangent  to  the  curve.    And  as  every  value  of  x  numerically  greater 
than  a,  positive  or  negative,  gives  imaginary  values  for  y,  there 
are  no  points  of  the  curve  without  the  tangents  at  A  and  B. 
y  =   0     gives 


a:  =   =h  a  =   CB  or  CA ; 

and  since  the  line  AB  bisects  a  system  of  chords  perpendicular  to 
it,  it  is  an  axis  of  the  curve,  Art.  (85),  and  A  and  B  are  its  vertices. 
X  =  0     gives 

y  =   ±  6  =   CD  or  CD'. 

Any  number  of  other  points  of  the  curve  may  be  constructed 
by  assigning  values  to  x  in  equation  (l),  deducing  and  ccnstruct- 
ing  the  coiTcsponding  values  of  y,  and  the  curve  in  form  and  po- 
sition will  be  as  in  the  last  figure. 

If  equation  (e)  be  solved  with  reference  to  .-r,  we  have 


^  =   =t  ^  V6«  -  2/S 


INDETERMINATE    GEOMETRY. 


133 


from  which  it  may  be  shown  as  above,  that  the  curvo  is  symme- 
trical with  respect  to  the  axis  of  Y,  and  does  not  extend  beyond 
the  tangents  at  D  and  D',  and  that  the  line  DD'  is  an  axis  of  the 
curve. 

The  definite  portion  of  the  line  AB,  included  within  the  ellipse!, 
is  called  the  transverse  axis,  and  the  portion  DD',  the  conjugate 
axis  ;  the  transverse  axis  being  the  longest  of  the  two. 

The  point  C,  in  which  the  axes  intersect,  is  the  centre  of  the 
ellipse. 

The  vertices  of  the  transverse  axis  are  also  called  the  vertices  of 
the  curve. 

Equation  (e)  is  called  the  equation  of  the  ellipse  referred  to  ii& 
centre  and  axes  ;  in  which  a  represents  the  semi-transverse,  and  h 
the  semi-conjugate  axis. 


107.     If  we  solve  equation  {h),  Art.  (105),  with  reference  to  y, 
we  have 


y'  =  ^  (^'  -  «■% 


Vl 


in  which  every  value  of  x  numerically  less  than  a,  positive  or 
negative,  gives  imaginary  values 
of  y :  Hence,  C  being  the  ori- 
gin, CX  and  CY  the  axes  of  co- 
ordinates, and  CA  and  CB  each 
numerically  equal  to  a,  there- 
are  no  points  of  the  curve  be- 
tween A  and  B. 

ic  =    -f  a     or     —   cr,     gives 

y  =   ±  0; 

hence,  the  ordinates  at  the  points  A  and  B,  when  produced,  are 
tangent  to  the  curve.     Every  value  of  x  greater  than  «,  positive  or 


134        '  INDETERMINATE    GEOMETRY. 

negative,   gives  two  real  values  of  y  equal  with  contrary  signs  ; 
hence,  the  curve  is  continuous  and  extends  to  infinity  in  both  di- 
rections beyond  the  points  A  and  B,  and  is  symmetrical  with  re- 
spect to  the  axis  of  X. 
y  =   0     gives 

a;  =   db  a  =  CA  or  CB, 

and  since  the  line  BA  produced,  bisects  a  system  of  chords  perpen- 
dicular to  it,  it  is  an  axis,  and  the  definite  portion  BA  =  2a, 
included  between  the  points  A  and  B,  is  called  the  transverse  axis 
of  the  curve,  the  points  A  and  B  being  its  vertices  or  the  vertices 
of  the  curve. 
X  =  0     gives 

,f  =   -  b\  y  =   rt  J  V  -   1; 

hence,  the  curve  does  not  intersect  the  axis  of  Y. 

A  sufficient  number  of  other  points  being  constructed  from  the 
equation,  the  curve  may  be  drawn  as  in  the  figure,  the  two 
branches  being  equal,  since  values  of  x  which  are  numerically 
equal  with  contrary  signs,  as  CP  and  CP'  give  the  same  values 
for  y. 

If  equation  (A)  be  solved  with  reference  to  ar,  we  have 

^  =  ^l  Vy^  +  6«, 

in  wh-<.  h  every  value  of  y  gives  two'  real  values  of  x,  equal  with 
contraiy  signs  ;  hence,  the  line  CY  is  an  axis  of  the  curve.  This 
line,  as  seen  above,  does  not  cut  the  hyperbola,  but  if  we  lay  off  on 
it  from  C,  distances  above  and  below  each  equal  to  6,  the  portion 
DD'  =  26  is  called  the  conjugate  axis^  the  point  C  being  the 
centre  of  the  hyperbola. 

Equation  (/?)  is  called  the  equation  of  the  hyperbola  referred  to 
its  centre  and  axes,  in  which  a  and  b  represent  the  semi-axes. 


INDETERMINATE    GEOMETRY. 


136 


108.  If  in  equations  (e)  and  (A),  and,  in  general,  in  the  equa- 
tion of  any  curve,  we  change  x  into  y 
and  y  into  x,  the  effect  is  to  change  the 
linO'  which  at  first  is  regarded  as  the 
axis  of  X,  into  the  axis  of  Y  and  the 
converse ;  or  if  the  curve  is  symmetri- 
cal with  respect  to  the  axes,  to  revolve 
it  90°  about  the  origin.  Thus  if  the 
equation 

represents  the  elHpse  as  indicated  by  the  full  line,  the  equation 

a2.T2  +  b^y^  =  a262, 
will  represent  it  as  indicated  by  the  broken  line. 


109.     If  a  point  is  on  the  ellipse,  its  co-ordinates  must  satisfy 
the  equation  of  the  ellipse.  Art.  (23),  and  we  must  have 

a^y^  +  ¥x^   -  a^^  =   0. 

If  the  point  is  without  the  ellipse,  y  will  be  greater  than  the  cor- 
.esponding  ordinate  of  the  ellipse,  Art.  (37),  and  we  have 

oV*  4-  h^x^  —„a%^    >  0. 

If  it  is  within  the  ellipse 


110.     The   corresponding    conditions   for   the    hyperbola,    by 
changing,  in  the  above,  b^  into  —  5',  Art.  (105),  will  be 

a^yi  _   b^x^  +  a^*  >  0. 
aV«  -  b^x^  +  a^b*  <  0. 


136 


INDETERMINATE    GEOMETRY. 


111.     If    a  =  6,     tlie  axes  of  the  ellipse  are  equal,  and  equa- 
tion {e)  becomes  * 

2,2  +  a;2  =:  a2, 

which  is  the  equation  of  a  circle,  the  radius  of  which  is  equal  to 
either  semi-axis.  Art.  (35). 


1]2.     Under  the  same  supposition,  equation  {ji)  becomes 

y2     _     a;2     —      _     ^2, 

and  the  curve  is  called  an  equilateral  hyperbola. 


113.     If  through  the   centre  of  an   ellipse  any  right  line  be 
WiT  drawn,  its  equation  referred  to  the 

axes  CX  and  CY,  will  be 

y  =  d'x (1,) 

^      in  which  d'  represents  the  tangent 
of  the  angle  which  the  line  makes 
with  CX,  Art.  (24). 
Combining  this  with  equation  (e),  by  substituting  for  y^  its  yalue 
rf'*a:',  we  obtain 

whence,  for  the  abscissas  of  the  points  of  intersection.  Art.   (27), 
we  have 


=  w; 


a%^ 


and  by  the  substitution  of  this  in  equation  (1), 

y  =   d=  a'  \ / . 

^  d'^a^  -f-  &» 


INDETERMINATE    GEOMETRY. 


137 


Since  these  values  of  x  and  y  are  real  for  every  value  of  d',  it 
follows  that  whatever  be  the  position  of  the  line  CM,  it  will  inter- 
sect the  ellipse  in  two  points;  and  since  the  co-ordinates  of  these 
points  are  equal  with  contrary  signs,  they  will  be  on  opposite  sides 
of  the  origin,  and  at  equal  distances  from  it,  as  at  M  and  M'. 
Hence,  every  straight  line  passing  through  the  centre  of  an  ellipse 
and  terminated  by  the  curve  is  bisected  at  the  centre. 


114.     If  in  the  above  expressions  we  put    —  b^  for  &*,  the  cor- 
responding values  for  the  hyperbola  are 


which  are  real,  whenever 

C?'2a2    -    62   <    0, 


or 


d'  <^, 


that  is,  whenever  d'^  either  positive  or  negative,  is  numerically  less 
than    —  ,  the  line  will  cut  the  hyperbola  in  two  points  and  be  bi 


sected  at  the  centre.     If 


d'   = 


the  values  are  both  infinite,  and  the  points  of  intersection  are  at 
an  infinite  distance  from  C.     If 

d-  >  1, 


the  values  are  imaginary,  and  the 
line  will  not  intersect  the  curve. 
Hence,  if  at  the  point  A,  we  erect 
the  perpendiculars  AE  and  AE', 
each  equal  to  6,  and  draw  the 
lines  CE  and  CE',  these  lines  will 
just  limit  the  curve,  since 


138  INDETERMINATE    GEOMETRY. 

tang  ACE  =  d'   .=  —  .=  L 
*'  CA         a 


115.    If  we  multiply  both  members  of  the  expression  p  z=  —  , 

a 


Art.  (105),  by  2,  we  have 


2i>  =  — , 
a 


which  as  in  the  parabola,  Art.  (84),  is  called  the  parameter,  and 
gives  the  proportion 

a   :   b   ::   2b   :   2p,         or         2a  :    2h   : :   26  :   2p. 

Hence,  the  parameter  of  the  ellipse  or  kype^'bol^i  is  a  third  pro- 
portional  to  the  ti'ansverse  and  conjugate  axes. 

116.     If  in  equation  (e),  we  substitute  for  y  the  expression    _, 

a 
we  find 

a;2  =  a«  —  b%  or         x  =   dz   Va^  —  b^\ 

and  conversely,  if  either  of  these  values  be  substituted  for  x,  we 
shall  find 

2/  =  ±  — ; 

a 

from  which  we  see,  that  there  are  two  points  on  the  transverse 
axis  of  the  ellipse,  at  which,  if  an  ordinate  be  drawn,  it  will  be 
equal  to  one  half  the  parameter  of  the  curve ;  hence,  the  double 
ordinate^  or  the  chord  perpendicular  to  the  transverse  axis,  at  each 
of  ihtse  points,  is  equal  to  the  parameter  of  the  curve. 


INDETERMINATE    GEOMETRY. 


139 


These  points  are  called  the  foci  of  the  ellipse,  and  may  be  con- 
structed thus  :  With  either  extremity  of 
the  conjugate  axis  as  a  centre,  and  the 
semi-transverse  axis  as  a  radius,  describe  an 
arc,  the  points  in  which  this  arc  cuts  the 
transverse  axis  will  be  the  foci.  For  in 
the  right  angled  triangle  DCF  or  DCF',  we  have,  Ait.  (4), 


CF*  =  CF'3  =  a*  -  6^         CF  =  CF'  =  ±   V^ 


h\ 


117.     For  the  hyperbola,  the  values  of  ar,  in  the  preceding 
article,  become 


=   rfc   Va^  +  h\ 


.(1), 


either  of  which  substituted  in  equation  (A),  will  give 

y      =      do      —y 


and  the  points  determined  on  the  transverse  axis,  by  laying  off  the 
above  values  of  x  arc  the  foci  of  the  hyperbola,  and  may  be  con- 
structed thus  :  At  either  vertex  of 
the  hyperbola  erect  a  perpendicu- 
lar equal  to  h  ;  join  its  extremity 
with  the  centre  ;  with  the  last  hno 
CE,  as  a  radius,  and  with  the 
point  C  as  a  centre,  describe  an  arc  ;  the  points  in  which  this  arc 
outs  the  transverse  axis  produced,  will  be  the  foci.     For  we  have 


CF*  =  GE'  =  a'  -\-  h\ 


CF  =  CF  =   d=   Va2  +   b\ 


^ 


118.     The  distance  from  the  centre  to  either  focua  of  the  ellipse, 


140 


INDETERMINATE    GEOMETRY. 


divided  by  the  semi-transverse  axis,  is  called  the  eccentricity  of  the 
curve,  the  expression  for  which  is 


If   a  =  6,  this  reduces  to  0  ;  hence,  the  excentricity  of  a  circle 
is  nothing,  and  the  foci  are  at  the  centi'e. 


119.     The  expression  for  the  eccentricity  of  an  hyperbola,  is 


OL. 


which,  when  a  —  b,  Art.  (112),  reduces  to  a/2.* 


120.     If  wo  denote  the  distance  CF  by  c,  and  the  distance  from 
^  any  point  of  the  ellipse,  as  M,  to  the  focus 

F,  by  r,  the  general  expression  for  the 
^  square  of  this  distance  will  be,  Art.  (IV), 


in  which 


^^, 


CT  ^    3f 


(^  -  ^'Y  +  (y  -   '/)'» 


y  =   0; 


whence 

FM*  =  7'«  =t  (ar  —  cf  4   y\ 
and  this,  by  the  substitution  of  the  values 

*  Note. — As  the  eccentricitj  of  the  ellipse  is  always  less  tiinn  l,aiil 
that  of  the  hyperbola  greater  than  1,  it  follows  that  the  eccentricity  <;, 
the  parabola  is  equal  to  1. 


INDETERMINATE    GEOMETRY.  141 


becomes 


^»  =   a;2  —   lex   +  a8  —  -lx\ 


or 


r*  =  - 


a^   -   2ca^x  +  c^x^.^^9^-     f^^sC   ik 


and  extracting^lie  square  root, 


a^  —  ex                   ex  /^s 

r   -.=  _— ^  ^  a (1), 


Using  tlie  plus  sign -of  the  root  only,  as  we  require  merely  tlie  ei- 
jiression  for  the  length  of  FM. 

Since  CF'  =  —  c;  if  in  the  above  expression  (1),  we  put 
—  c  for  c,  we  shall  evidently  obtain  the  distance  F'M,  which  we 
denote  bv  r' ;  hence, 

'^  =  <^   +  - (2). 

a 

Adding  equations  (1)  and  (2),  member  to  member,  we  have 
r  ~\-  r'   =   2a ; 

hence,  the  sum  of  the  distances  from  any  point  of  the  curve  to  the 
two  foci  is  cq?ial  to  the  transverse  axisi 

This  remarkable  property  enables  us  to  define  an  ellipse  to  be,  a 
curve  such,  that  the  sum  of  the  distances,  from  any  point  to  two  fixed 
jwiuts,  is  always  equal  to  a  yiven  line. 

It  also  gives  the  following  construction,  of  the  curve  by  points, 

the  foci  and  transverse  axis  being  given.  \;,£ 

I>ivide  the  transverse  axis  into  any  two 

parts,  the  point  of  division  being  between        /     ^^.-^^  \   \ 

the  two  foci,  as  at  E;  wiili  oiui   part  EB       ^  F'  M  J<!  B 


!  i2  INDETERMINATE    GEOMETJIT. 

as  a  radius,  and  one  focus  F,  as  a  centre,  describe  an  arc ;  with  the 
other  part  AE,  as  a  radius,  and  the  other  focus  as  a  centre,  describe 
a  second  arc ;  the  points  of  intersection  of  these  arcs  will  be  points 
of  the  ellipse.     For  we  have 

FM  +  F'M  =  EB  +  AE  =  2a. 

The  curve  may  also  be  constructed  by  a  continuous  movement, 
thus  :  Take  a  thread,  in  length  equal  to  the  transverse  axis,  and 
fasten  an  end  at  each  focus ;  press  a  pencil  against  the  thread  so 
as  to  draw  it  tight ;  the  point  of  the  pencil  as  it  is  moved  around 
will  describe  the  ellipse ;  for  the  sum  of  the  distances,  fjom  this 
point  to  the  foci,  is  always  the  same  and  equal  to  the  transverse 
axis. 


^^A(C^     121.     If  F  and  F'  are   the  foci  of  the  hyperbola,  and  the  dis- 


7  ^  tances  FM  and  F'M  be  denoted  by  r  and  r',  we  may  deduce  ex- 
pressions for  them  from  ex- 
pressions (1)  and  (2)  of  the 
preceding  article,  by  chang- 

-M^ x_j^__j__  ing  ly^  into    — -  6*,   the  only 

effect  of  which    will  be  ta 


make    c  =  yo^-i-   V^    in- 


k 


stead  of     Va^--^,     and 
as  for  all  points  of  the  curve  x  must  be  greater  than  a.  Art.  (lOV), 

—   must  also  be  greater  than  a,  and  the  expression  for  the  nu- 
3 
m^rical  value  of    FM  =  r    will  be 


r^   -  -   « ^(3). 

a 

The  form  of  the  expression  for  r'  will  remain  unchanged  ;  hence, 

r'  =   -~   -f-  « W. 

,     a 


INDETERMINATE    GEOMETRY.  143 

Subtracting  the  first  of  these  from  the  second,  we  have 
r'  —  r  =   2a; 

hence,  the  difference  of  the  distances  from  amj  point  of  tlie  curve  to 
(he  iioofoci,  is  equal  to  the  transverse  axis  ;  and  the  hyperbola  may 
be  defined  to  be,  a  curve  such,  that  the  difference  of  the  distances 
from-  any  point  to  two  fixed  points  is  equal  to  a  given  line. 

The  curve  may  be  constructed  by  points  thus  :  With  one  focus 
F'  as  a  centre,  and  any  radius  F'E,  greater  than  the  distance  to 
the  farther  vertex,  describe  an  arc ;  with  the  other  focus  and  a 
radius  FM,  equal  to  the  first  radius  minus  the  transverse  axis,  de- 
scribe another  arc  ;  the  points  of  intersection  will  be  points  of  the 
curve.     For,  we  have 

F'M  —  FM  =   2a. 

It  may  also  be  constructed  by  a  continuous  movement.  Take  a 
rule  of  sufficient  length  as  F'L,  and  fasten  one  end  at  the  focus 
F' ;  at  the  other  end  of  the  rule  fasten  one  end  of  a  string  shorter 
than  the  rule  by  the  transverse  axis ;  fasten  the  other  end  of  the 
string  at  the  other  focus,  F  ;  press  a  pencil  against  the  string  and 
rule ;  as  the  rule  revolves  about  the  focus  F',  the  point  of  the 
pencil  will  describe  the  branch  AM.     For,  we  have 

PL  -•  2a  =--  FM  +  ML, 


or 


henco 


F'L  —  ML  -  FM  =  2rt 


F'M  —  FM  =  2a, 


By  placing  the  end  of  the  rule  at  F,  the  other  branch  may  be 
described. 


144  INDETERMINATE   CEOMETRY. 

122.  By  a  reference  to  equations  (1)  and  (2),  Art.  (120),  it  is 
seen  that  the  distance  from  any  point  of  the  curve  to  either  focus  is 
expressed  rationally  in  terms  of  its  abscissa. 

This  remarkable  property  of  the  foci  is  possessed  by  no  other 
points  in  the  plane  of  the  curve.  For,  if  there  is  any  other  point, 
let  its  co-ordinates  be  x'  and  y' ;  x  and  y  denoting  the  co-ordinates 
of  any  point  of  the  curve.  The  square  of  the  distance  from  x,  y, 
to  x',  y'.  Art.  (IV),  is 

I)^  =  {x  -  x'Y  +  {y  -  y'Y, 

or  squaring    x  —  x'     and     y  —  y',     and  substituting  for  y  its 
vahie 


a 


we  have 


1>2  =  — ^x^  —  2xx'  +  x'^  +  b^=P   2y' A  W^2  _  ^2    ,    ^/a 

a^  'a  T  y   • 

It  is  evident  that  the  value  for  D  can  not  be  rational,  in  terms 
of  X,  unless  the  term  containing  the  radical  disappears.  But  this 
can  not  be  unless  y'  j=  0,  that  is,  the  required  point  must  be 
on  the  axis  of  X.  Substituting  this  value  for  y',  D'^,  after  changing 
the  order  of  the  terms,  becomes 

D2  =  {h^  -f  x'^)  -  2xx'  +  ''''  ~  ^'  x\ 

a^ 

Now  no  value  of  x'  can  make  this  expression  a  perfect  square 
unless  it  makes  the  first  and  last  terms  perfect  squares,  and  twice 
the  square  root  of  their  product  equal  to  the  middle  term,  that  is, 
we  must  have 

i«   -f  x'^  =  m«,         ^^Lziilir*  r=  n\  —  2xx'  =  2mn, 


INDETERMINATE    GEOMETRY. 


145 


vi^  and  n^  being  two  perfect  squares.     From  the  last  expression 
we  havft 


!.«      — 


in  which,  substituting  the  value  of  n^  taken  from  the  second,  we 
have 


wi^  = 


aVa 


aa  _   63 

Substituting  this  in  the  first,  we  have 

5«   +  x'^  = 


a2  -   6»  ' 

which  can  be  satisfied  for  no  values  of  x',  except 

X'  =   ±   Va^  -  b%    ^\  \  ^  j 

the  abscissas  of  the  two  foci. 

In  a  similar  way  it  may  be  shown,  that  the  foci  of  the  hyperbola 
and  parabola  alone  possess  the  above  named  property. 


H 


123.     If  in  equation  (//-)  we  substitute  h  for  y,  w^e  deduce 

a;2   =   2a2,  x   =   a  V^', 

therefore,  the  abscissa  of  that  point  of  the  hyperbola,  whose  ordi- 
nate is  equal  to  the  semi-conjugate  axis,  is  equal  to  the  diagonal 
of  a  square,  the  side  of  which  is 
the  semi-transverse  axis.  Hence, 
the  curve  and  transverse  axis 
being  given,  the  conjugate  axis 
may  be  constructed  thus  :  At  the 
vertex  A,  erect  a  perpendicular  AR  =  a  ;  join  the  extremity, 
10 


I 


\^ 


.;M 


^.  0 


146  INDETERMINATE    GEOMETRY. 


with  the  centre  ;  with  C  as  a  centre,  and  CR  as  a  radius,  describe 
an  arc  cutting  the  transverse  axis  in  0,  at  which  erect  the  ordinate 
OM ;  it  will  be  equal  to  the  semi-conjugate  axis.     For  we  have 

CO  =  CR  =  CA  V2  =  aV2, 


124.    If  the  values 

a  a' 

Art.  (106),  be  substituted  in^eg^uation  (1)  of  the  same  articlet. 
giving  to  r**,  first  the  negative  and  then  the  positive  sign,  we  ob- 
tain the  two  equations 

y«  =  ^{2ax  -  x^) (1), 

y^  =^^^  (2a^   4-  ^'') (2), 

which  are  the  equations  of  the  ellipse  and  hyperbola  referred  to 
the  axis  and  principal  vertex  A.    See  figures  of  Arts.  (106),  (10*7). 


125.  Let  a;',  ?/',  and  x'\  y",  be  the  co-ordinates  of  any  two 
pomis  of  the  ellipse.  These  co-ordinates,  when  substituted  for  x 
and  y  in  equation  (e),  must  satisfy  it,  Art.  (23),  and  .give  the  two 
equations  of  condition 

or 

y«  =  -^(a«  -  x'^) (1),         y"'  =  ^{«*  -  ^"y 

a*  a* 

Dividing  the  first  by  the  second,  member  by  member,  we  have 


INDETERMINATE    (JEOMETRY. 


147 


^ 


y!%    _  ««  —  x'^    _   (a  +  x'){a   —  x') 

y^  ~  a2  —   x"'^  ~   (a  4-  x")(a  —  x")  ' 

whence,  we  deduce  the  proportion 

y'«  :   y"3   :  :    (a  +  a;')(a  —  x')    :    (a   +  x"){a  -  ar" 

But 

a  -f  a/  =  AP,  a  —  a;'  =  PB,                        J>      ' 

a  +  a;"  =  AP',  a  —  x"  =  PT>. 


•i  1 


^ 


T^Tj2 


Therefore 


MP*   :   M'P''   :  :  AP  X  PB   :   AP'  x  P'B ; 

that  is,  the  squares  of  the  ordinates  of  any  two  points  of  the  ellipse 
are  to  each  other  as  the  rectangles  of  the  segments  into  which  they 
divide  the  transverse  axis. 

For  the  circle,    a  =  b,   and  equation  (l)  reduces  to,  Art.  (36), 

7/'2  =  a2  -.  a:'«  =   (a  -f  x')(a  —  x'). 


126.     By  using  equation  (?i)  and  pursuing  the  same  method  as 
in  the  preceding  article,  we  shall  find  for  the  hyperbola 


y'«   :   y"»   :  :    {x'  +  a){x'  -a)  :  (x"   +  a){x" 


a\ 


or 


MP' 


M'F     :  AP   X   BP   :   AP'  x  BP', 


that  is,  the  squares  of  the  ordi- 
nates of  any  two  points  of  the 
hyjyerhola,  are  to  each  other  as  the 
rectangles,  of  the  distances  from 
the  foot  of  each  ordinate  to  the 
vertices  of  the  curve. 


A  i  £ 


148 


INDETERMINATE    GEOMETRT. 


121.    If  With  tlie  centre  of  the  eUipse  as  a  centre,  and  CA  =  a 
- '  'r-'.^^['  as  a  radius,  a  ch'cle  be  described,  its  equa- 

tion, Art.  (Ill),  may  be  put  under  the 
form 


Y2     =     rt* 


•(1), 


^  in  which  Y  represents  the  ordinate  of  any 

point  of  the  circle,  as  MT.     From  equation  (e)  we  have 


y'  =  ^K 


x^). 


•(2), 


in  which,  if  x  have  the  same  value  as  in  equation  (1),  y  will  rcpre« 
sent  the  ordinate  MP,  of  the  eHipse.  Dividing  equation  (2)  by 
(1),  member  by  member,  we  have 


Y8 


whence 


Y    :    7/   :  :    a    :    b, 


that  is,  if  a  circle  be  described  on  the  transverse  axis  of  an  ellipse, 
any  ordinate  of  the  circle  will  hs  to  the  corresponding  ordinate  of  the 
ellipse  as  the  semi-transverse  to  the  semi-conjiujate  axis. 

If  with  C  as  a  centre,  and     CD  =■  h     as  a  radius,  a  circle  bo 
described,  its  equation  may  be  put  under  the  form 


J2       _        y^ 


(3), 


in  which  X  represents  the  abscissa  of  any  point  of  the  circle  aa 
RN'.  If  we  obtain  tlie  value  of  x^  from  equation  (e)  and  divide 
by  equation  (3),  we  may  deduce  the  proportion 

\    '.    X    '.  '.    h    \    a, 

that  is,  if  a  circle  be  described  on  the  conjugate  axis  of  an  ellipse, 


hJdeterminate  geometry. 


149 


any  abscissa  of  the  circle  will  be  to  the  corresponding  abscissa  of  the 
ellipse  as  the  semi-conjugate  to  the  semi-transverse  axis. 

From  the  first  of  the  above  proportions,  it  appears  that  the  or- 
dinate of  any  point  of  the  circle  described  on  the  transverse  axis, 
is  greater  than  the  corresponding  ordinate  of  the  ellipse  ;  hence, 
all  the  points  of  this  circle  are  ^Yithout  the  ellipse,  except  the  ver- 
tices A  and  B. 

From  the  second  proportion,  it  also  appears  that  every  point  of 
the  circle  described  on  the  conjugate  axis  is  within  the  ellipse,  ex- 
cept the  vertices  D  and  D'. 

We  also  conclude,  that  of  all  straight  lines,  passing  through  the 
centre,  and  terminating  in  the  ellij^se,  the  transverse  axis  is  tlie 
longest,  and  the  conjugate  the  shortest. 

Upon  the  above  properties,  the  following  constructions  of  the 
ellipse  depend. 

First.  On  each  of  the  axes  as  a  diameter,  describe  a  circle  ;  at 
any  point  of  the  transverse  axis,  as  P, 
erect  a  perpendicular  and  produce  it,  till 
it  meets  the  outer  circle  in  M' ;  join  this 
point  with  the  centre  by  the  line  M'C ; 
from  the  point  R,  where  this  line  meets 
the  inner  circle,  draw  a  line  parallel  to  the  transverse  axis,  the 
[K)int  in  which  it  meets  the  perpendicular  will  be  a  point  of  the 
ellipse.     For,  we  have 


M'P   :   MP   :  :   M'C   :   RC   :  :   a  :   6. 


Second.  Take  a  rule  MO,  in  length  equal  to  the  semi-transvetse 
axis  ;  from  the  extremity  M,  lay  oflF  MS 
equal  to  the  semi-conjugate  axis  ;  move 
the  rule  so  that  the  extremity  O  and 
the  point  of  division  S  shall  remain,  the 
first  on  the  conjugate  and  the  second  on 
the  transverse  axis ;  the  point  of  a  pen- 
cil at  M,  will  describe  the  ellipse.     For,  draw  OP'  parallel  to  CB, 


/"' 

-— 

'~y 

vf 

/^ 

— 

-77 

WA 

r 

//' 

"^ 

A. 

C 

A 

0 

J 

^'       i' 

6        iP^/» 


150 


INDETERMINATE    GEOMETRY. 


dutil  it  meets  the  produced  ordinate  MP  in  P' ;  join  M'C,  then 
the  two  equal  right  angled  triangles  M'CP  and  MOP'  give 

MP'  =  M'P, 

and  the  similar  triangles  MPS  and  MP'O  give 

MP   :  MP  ::  MO  :  MS  ;:  a  :  h. 


128.     Let  x"^  y",  be  the  co-ordinates  of  any  point  of  the  ellipse, 
as  M,  and  through  this  point  conceive  any  straight  line  to  be 
^j.  2J  drawn  ;    its  equation  will 

be  of  the  form 

y-y"  =  d{x  -  x")...{l\ 

in  which  d  is  undetermined.  Since  the  given  point  is  on  the 
curve,  its  co-ordinates  must  satisfy  equation  (c),  and  give  the  con- 
dition 

Subtracting  this,  member  by  member,   from  equation  (c),  we 
have 

w 

,/.'(y  4-  y/)(y  —  y")   +  h\x  +  x"){x  —  x")  =  0. 

Combining  this  with  equation  (1),  by  substituting  the  value  of 
V  —  y"     taken  from  equation  (1),  we  obtain 

[rfa«(y  +  y")   +  h\x   -f  x")\{x  —  x")   =   0, 

in  which  x  and  y  are  the  co-ordinates  of  all  the  points  common  to 
tlio  right  line  and  curve.  This  equation  being  of  the  second  de- 
S^veo,  there  are  two  such  points,  and  only  two.     These  points  may 


INDETERMINATE    GEOMETRY.  151 

be  determined   by   placing   the   factors,  separately,   equal  to  0. 
Placins: 


X  —  x'\ 


X  —  x"  =  0, 

we  have 

which  in  equation  (1),  gives 

y 

=  y", 

and  these  values  evidently  belong  to  the  given  point  M.     Placing 
the  other  factor  equal  to  0,  we  have 

da\y   4-  y")   +   h\x   +  x")   =   0 (2), 

in  which  x  and  y  must  be  the  co-ordinates  of  the  second  point  of 
intersection  M'. 

If  now  the  right  line  be  revolved  about  the  point  M,  until  the 
point  M'  coincides  with  M,  the  secant  line  will  become  a  tangent ; 
X  and  y,  in  equation  (2),  will  become  equal  to  x"  and  y",  and  the 
equation  reduce  to 

2dahj"  +  26V  =  0 ;  whence         d  =    —  lf_  . 

ay 

Substituting  this  value  of  c?  in  equation  (1),  we  have 

h^x" 

y  -  y"  = — (^  -  x'%       ' 

ahj" 

which,  since     ahj"^  +  ¥x"^  =  a%^,    reduces  to 

ahjy"   +   h'^xx"   —   a%^ (3), 

for  the  equation  of  a  tangent  liney  to  the  ellipse  at  a  given  ])oint. 
If    a  =  6,     the  above  equation  reduces  to 

yy"  +  xx"  =  «S 
for  the  tangent  line  to  the  circle  whose  radius  is  cu 


152 


INDETERMINATE    GEOMETRY. 


129.  If  we  multiply  both  members  of  equation  (3),  preceding 
article,  by  2,  and  subtract  the  result,  member  by  member,  from 
the  equation 

we  have 

ahj"^  —   2ahjy"   +   h'^z"^  —    2h^xx"   —   —   a%^. 
Adding     a^y^  +  h-x"^,     to  both  members,  we  have 

a\y"   —  7jY  +  h''{x"  —  xY  =  a^y"^  -f   h^x^  —  a"b\ 
The  first  member  is  the  sum  of  two  perfect  squares,  hence 

aV  +  ^"^^"^  —  (^^f  - 
is   positive   for   all  values   of  x  and  y,   except     x  =  x"     and 

y  =  y"' 

All  points  of  the  tangent^  except  the  point  of  contact,  are  there- 
fore without  the  ellipse,  Art.  (109). 


130.     If  in  equation  (3),  Art.  (128),  we  make     y  =  0,     we 

find 


a:  =  ^  =  CT,       -ii 

X" 


^  for  the  distance  from  C,  to 

the  point  in  which  the  tangent  cuts  the  transverse  axis.     If  from 
this  we  subtract  the  distance     CP  =  x",     we  have 

CT  —  CP  =  PT  =  ^^ x"  =  ^ ^, 


which  is  the  subtangent.  Art.  (92).  Tliis  expression  for  the  sub- 
tangent,  being  independent  of  the  conjugate  axis,  will  be  the  same 
for  all  ellipses  having  the  same  transverse  axis,  and  the  points  of 
contact  in  the  same  perpendicular  to  this  axis.     Hence,  if  it  be 


INDETERMINATE    GEOMETRY. 


153 


required  to  draw  a  tangent  to  an  ellipse  at  a  given  point  as  M  :  On 
the  transverse  axis  describe  a  circle  ;  through  the  given  point  draw 
a  perpendicular  to  the  axis  and  produce  it  until  it  meets  the  circle 
at  M' ;  at  this  point  draw  a  tangent  to  the  circle,  and  connect  the 
point  T,  in  which  this  tangent  cuts  the  axis  produced,  with  the 
given  point ;  this  line  will  be  the  required  tangent. 

In  a  similar  way,  we  may  find  the  distance  cut  off  by  the  tan- 
gent on  the  conjugate  axis  produced,  and  the  expression  for  the 
subtangent  on  this  axis. 


h 


'131.     If  in  equation  (S),  Art.  (128j,  Ave  change  5*  into  —  b%  it 
becomes 


ahjy" 


for  the  equation  of  a  tangent  to  the  hyperbola  at  a  given  point. 


X^'^  =  CT (1), 


132.     If  in  the  last  equation  we  make     y  =  0,    we  find 

N  J" 

and   subtracting    this   from  ^ 

CP  =  x"^    we  have 

CP  _  CT  =  PT  = 


x'''^  -  a^ 


\ 


for  the  subtangent  of  the  hyperbola. 


133.     Let  MT  be  a  tangent  at  any  point  M,  of  the  ellipse,  the 
co-ordinates  of  this  point  being  x"  j,, 

and   y"\  draw  the   lines   MF    and 
MF'  to  the  foci.     In  Art.  (120),  we         / 
have  found  / 


*"          "       T*-        "" 

164                                         INDETERMINATE    GEOMETRY. 

MF'    -  a   +  ^"^'^ 

MF  =  o  - 

cr" 

> 

a 

a 

or 

MF='^'+'^", 

MP  =  "'-, 

cxf' , 

hence 

MF'   ;  MF  :  :   a«  4-  c^'   :   «'  —  ex!' (i). 

If  to  the  expression    CT  =  ^,    Art.  (130),  we  add  CF'  =  c, 
and  from  it  subtract     CF  =  c,     we  have 


,  *  *     — , 

Jt"  x" 


hence 

F'T  :  FT  :  :  a«  +  ex"  :  a^  -  c:r", 
and  since  the  last  terms  of  this  proportion  are  the  same  as  (1), 
F'T  :  FT  :  :  MF'  :  MF. 
Through  F  draw  FO  parallel  to  MF',  then 
F'T  :  FT  :  :  MF'  :  FO ; 
hence     FO  =  FM,     and  the  angle 

FMO  =  FOM  =  F'MT'. 

Therefore,  if  from  the  point  of  contact  of  a  tangent  to  an  ellipse^ 
two  lines  he  drawn  to  the  foci^  these  lines,  will  make  equal  angles 
with  the  tangent. 

Tliis  property  enables  us  to  make  the  following  constructions. 

First.  To  draw  a  tangent  to  an  ellipse  at  a  given  point. 
Join  tlie  point  wth  the  foci ;  produce  the  line  F'M,  drawn  to  one 


INDETERMINATE    GEOMETRY/  155 


focus,  until  it  is  equal  to  tlie  trans- 
verse axis;    join    its    extremity    B', 
with  tlie  other   focus;   through  the 
given  point  draw  a  line  perpendicu-        ^— jH? 
lar  to  the  last  line ;  it  will  be  the  tangent.     For, 

F'M  +  MB'  =  2a  =  F'M  +   MF, 

hence  MB'  =  MF,  the  triangle  MFB'  is  isosceles,  and  the 
angle 

B'MT  =  F'MN  =  FMT. 

"^^  Second,  To  draw  a  tangent  to  an  ellipse  from  a  point  without 
the  curve. 

AVith  either  focus  F',  as  a  centre,  and  radius  equal  to  the  trans- 
verse axis,  describe  an  arc ;  with  the  given  point  N,  as  a  centre, 
and  radius  equal  to  the  distance  to  the  other  focus,  describe  ano- 
ther arc ;  join  their  point  of  intersection  B',  with  the  first  focus  ; 
the  point  M,  in  which  this  line  intersects  the  ellipse,  will  be  the 
point  of  contact,  which  being  joined  with  the  given  point  will 
o-ive  the  tanofent.     For 

NF  =  NB'  and         MF  =  Mp' ; 

hence  the  line  NM,  having  two  points  at  equal  distances  from  F 
and  B',  is  perpendicular  to  FB'  at  its  middle  point  and  bisects  the 
angle  FMB'.  Since  the  two  arcs  above  described  intersect  in  two 
points,  there  will  be  ^wo  tangents. 

Let  the  co-ordinates  of  the  given  point  be  x'  and  y'.  Since  it 
is  on  the  tangent,  we  must  have  the  condition.  Art.  (23), 

a^y'yit   4.  lix'x"  —  a%^ (2), 

and  since  the  point  of  contact  is  on  the  ellipse,  we  also  have 

aY'2   +  &V'2  =  a%^ (3). 

The  combination  of  these  equations  will  give  two  values  ofz" 


150  INDETERMINATE    GEOMETRY 

and  two  corresponding  of  y'\  Art.  (93),  which  will  be  of  the  form 

y"   =   7n  zt  n  VaY^   +   b^x'^   —   a%'\ 

and  these  values  will  be  real,  equal,  or  imaginary  as  the  given 
point  is  without,  on,  or  within  the  ellipse.  Art.  (109),  In  the  first 
case  there  will  be  two  tangents,  in  the  second  but  one,  and  in  the 
third  none. 


134,     Let  MT  be  a  tangent  to  the  hyperbola  at  any  point, 

and  MF'  and  MF  lines  drawn 
to  the  foci.  In  Art.  (121), 
we  have  found 


MF     r= 


MF  = 


ex" 

+ 

a« 

a 

ex" 

— 

a« 

If  to     c  =  CF'  =  CF,     we  add  the  expression     CT  =  _, 

x" 

Art.  (132),  and  then  subtract  it,  we  shall  have 


F'T  = 


ex"   +  a^ 


FT  .-= 


ex"  —  a* 


Ileuce,  as  in  the  preceding  article,  we  deduce 

F'T  :  FT  :  :  MF'  :  MF, 

/    that  is,  the  tangent  MT  divides  the  base  of  the  triangle  MF'F  into 

\    two  segments  proportional  to  the  adjacent  sides,  it  therefore  bi- 

(     sects  the  angle  F'MF,  at  the  vertex.     Therefore,  if  from  the  2^oint 

of  contact  of  a  tangent  to  an  hyperbola,  two  lines  be  draiim  to  the 

foeiy  these  lines  will  make  equal  angles  inth  the  tangent. 

This  property  enables  us  to  make  the  following  constructions. 


INDETERMINATE    GEOMETRY.  157 

I'irst.  To  draw  a  tangent  to  an  hyperbola  at  a  given  point. 
Join  the  point  M,  with  the 
foci ;    with  the  point   as  a 
centre  and  the  distance  to 
the  nearest  focus  as  a  radius, 

describe  an  arc  cuttinrr  the  ^ 

hne  drawn  to  the  farthest  / 

focus  in  A' ;  join  this  point  with  the  first  focus  and  through  the 
given  point  draw  a  hne  perpendicuhir  to  this  last  hne  ;  it  will  be 
the  tangent.  For  the  triangle  MFA'  is  isosceles  ;  hence,  the  per- 
pendicular MT  bisects  the  angle  F'MF. 

Second.  To  draw  a  tangent  to  an  hyperbola  from  a  point  with- 
out the  curve. 

The  construction  and  explanation  of  this  are  the  same  as  for  the 
ellipse. 

If,  as  in  the  ellipse,  the  co-ordinates  of  the  given  point  be  de- 
noted by  x'  and  ?/',  we  shall  have  for  the  hyperbola  the  two  equa- 
tions of  condition 


h^x'x"    =     -    ^2^2 


the  combination  of  which,  will  give  two  values  of  x"  and  y",  which 
will  1tre-of4he  form 

y"   =   7,1   dr   n  ^ a^y<^   —    b'-x'^   -f  a^^, 

and  there  will  be  two  tangents,  one,  or  none,  as  the  given  point  w 
without,  on,  or  within  the  hyperbola. 


/  '  inor 


135.     The  general  form  of  the  equation  of  a  straight  line  pass- 
ing through  the  point  B  is.   Art.  ^--^^ 
(29), 

y  —  y'  =  c(x  —  x'), 


/:-r^,-  ^ 


ir.S  INDETERMINATE    GEOMETRY. 

in  which,  for  this  particular  case,  we  must  have 
y'  =   0,  x'   =  a, 

which  gives  for  the  equation  of  BM, 

y  —   c{x   -^  a). 
For  the  equation  of  the  right  line  passing  through  A,  for  which 
y  =  0,  X    =   —  a, 

we  have 

y  =  C  (a;  4-  a). 

Combining  these  equations  by  multiplication,  we  have 

y*  =  cc'  (x*  —  a*), 

which  must  be  satisfied  by  x  and  y,  the  co-ordinates  of  the  point 
of  intersection  of  the  two  lines,  Art.  (27).  If  this  point  is  on 
the  ellipse,  x  and  ?/  must  also  satisfy  the  equation  of  the  ellipse, 
and  we  must  have 

y*  =  %  (a«  _  ^=)  =   _  ^  (X'  -  J). 

a*  a*  f 

Equating  these  values  of  y*,  and  omitting  the  common  factor 
x*  —  a*,     we  have 

<^''  = T (1). 

a* 

for  the  equation  of  condition  that  the  lines  shall  intersect  on  the 
ellipse. 

The  Hnes  when  subjected  to  this  condition  are  called  supple^ 
mentary  chords  ;  and,  in  general,  supplementary  chords  of  a  ctirve 
are  straight  lines  dro,wnfrom  the  extremiVus  of  a  diameter  and  in- 
tersecting on  the  curve. 

Since  cand  c'  are  indeterminate  in  the  above  equation  of  condi- 


INDETERMINATE    GKOMETUY.  159 

tion,  an  infinite  number  of  supplementary  chords  can  be  drawn, 
and  if  any  value  be  assigned  to  either  c  or  c',  the  other  becomes 
known  and  the  position  of  the  corresponding  chord  will  be  de- 
termined. 

If  c  =  0,  (/  will  be  00 ;  or  if  c'  =  0,  c  =  oo ;  that 
is,  if  either  chord  coincides  with  the  transverse  axis,  the  other  will 
be  perpendicular  to  it. 

If  either  c  or  c'  is  positive,  the  other  must  be  negative  ;  that  is, 
if  one  chord  makes  an  acute  angle  with  the  transverse  axis,  the 
other  will  make  an  obtuse,  and  the  reverse. 

If    a  =  b,     the  condition  (1)  reduces  to 

cc'  =   —   1,  or  cc'    +   1   =  0  ; 

hence,  Art.  (28),  the  supplementary  chords  of  a  circle  are  perpen- 
dicular to  each  other. 

The  expression  for  the  tangent  of  the  angle  AMB  is,  Art.  (28), 

yr  C    —    c' 

tang  V  z: 


1    +  cc' 


But  since  c  is  the  tangent  of  the  obtuse  angle  MBX,  it  is  essen- 
tially negative  and  may  be  placed     =   —  c".     Substituting  this, 

and  also      cc'  =   —  — ,     the  above  expression  becomes 


tang  V  =    -  (^"   ^  ^') 
1     _     ^ 

/i8 


which  is  essentially  negative  for  all  values  of  c  and  c  ;  hence  the 
supplementary  chords,  drawn  from  the  extremities  of  the  trans- 
verse axis  of  an  erii[.se,  make  an  obtuse  angle  with  each  other. 

As  the  angle  V  is  obtuse,  it  will  be  the  greatest  when  its  tan- 
gent is  numerically  the  least ;  and  since  the  denominator  of  the 
above  expression  is  constant,  it  will  be  the  least  when  the  numera- 


160  INDETERMINATE    GEOMETRY. 

tor  is  tlie  least.  But  tlie  product  of  d  and  c"  being  constant,  their 
sum  c"  +  C,  will  be  the  least  when  the  factors  are  equal,* 
that  is,  when 

c"  =  c',         or         —  c  =  c', 

in  which  case,  the  angles  are  supplements  of  each  other  and  the 
chords  are  drawn  to  the  extremity  of  the  conjugate  axis  D,  making 
theande    DAC  =  DEC. 


136.  If  we  put  ~  6^  for  W'  in  condition  (1)  of  the  preceding 
article,  we  obtain 

cd    =  —, 

for  the  equation  of  condition  for  supplementary  chords  drawn  from 
the  extremities  of  the  transverse  axis  of  the  hyperbola. 

As  in  the  ellipse,  an  infinite  number  of  chords  may  be  drawn, 
and  if  either  c  or  c'  is  positive  or  negative,  the  other  must  have 
the  same  sign ;  that  is,  both  angles  are  at  the  same  time  acute,  or 
both  obtuse. 

If  a  =  b,     the  above  equation  becomes 


* 
then 

NOTE.- 

— To  prove  this,  let  s  represent  their  sum  and  d  their  differenco, 
£  +  1  =  the  greater,           f.  _  ^  =  the  less, 

and 

fl  _  ^  =  the  product  = 

P. 

or 

4             ^4' 

from  which  we  see  that  s^  or  s  will  be  the  least  when    d  =  0,    or  the  two 
factors  are  equal. 


INDETERMINATE    GEOMETRY. 


ici 


or 


hence,  in  the  equilateral  hyperbola  the  two  angles  are  complements 
of  each  other. 

The  expression  for  the  tangent  of  the  angle  BMA,  is 


tang  V  = 


1   +  cc' 


which  is  essentially  positive, 

since  c  is  always  gi'eater  than 

c' ;  hence,  the  supplementary  chords  make  an  acute  angle  with 

each  other,  and  this  angle  increases  as  c  increases,  until  the  chord 

AM  becomes  perpendicular  to  the  transverse  axis  at  the  vertex  A, 

when  the  angle  is  the  greatest  possible  and  equal  to  90°. 


137.    If  a  right  line  be  drawn  through  the  centre  of  the  ellipse, 
its  equation  will  be  jv-^ 


and  if  it  pass  through  the  point  of       ^| 
contact  of  a  tangent,  we  shall  have 
the  condition 


y"  =  d'x",  or  d'  =  1-. 

X" 


ci^ 


Multipljdng  this,  member  by  member,  by  the  expression  for  d, 
Art,  (128,)  we  have 


dd'  = 


b* 


.(1), 


the  same  expression  as  that  found  in  Art.  (135)  for  cc' ;  hence 
11 


162  INDETERMINATE    GEOMETRT 

rr.'    =    dd\ 

in  which,  if  c  =z  d,  c'  will  be  equal  to  c?',  and  if  c'  =  d', 
c  =  d. 

Therefore,  if  one  of  the  supplementary  chords  of  an  ellipse  is 
parallel  to  a  tangent^  the  other  will  he  parallel  to  a  line  joining  the 
point  of  contact  and  the  centre^  and  the  converse. 

Upon  this  property  the  following  constructions  depend. 

First.  To  draw  a  tangent  to  an  eUipse  at  a  given  point.  From 
tJie  point,  draw  a  line,  MC,  to  the  centre ;  from  one  extremity  of 
the  transverse  axis  draw  a  chord,  AO,  parallel  to  this  line  ;  draw 
the  supplement  BO,  of  this  chord,  and  at  the  given  point,  draw  a 
line  parallel  to  this  supplement,  it  will  be  the  required  tangent. 

Second.  To  draw  a  tangent  to  the  ellipse  parallel  to  a  given 
line. 

From  one  extremity  of  the  transverse  axis,  draw  a  chord,  BO, 
parallel  to  the  given  line  NS ;  draw  the  supplement  of  this 
chord  AO  ;  parallel  to  which  draw  a  line,  CM,  through  the 
centre  ;  at  the  points  in  which  this  line  intersects  the  curve,  draw 
lines  parallel  to  the  given  line,  they  will  be  the  required  tangents. 


138.     By  changing  h"^  into  —  6^,  in  the  expression  for  c?,  Art. 
(128),  it  becomes  the  tangent  of  the  angle  made  with  the  transverse 

axis  by  a  tangent  to  the 
hyperbola,  and  by  using 
this  ex'pression  with  the 
equation  of  condition 


y"  =  d'x", 

we  have  a  similar  discussion,  and  deduce  the  same  properties  of 
supplementary  chords,  and  the  same  constructions  for  tangent  lines 
as  in  the  ellipse,  as  indicated  in  the  figure. 

It  will  evidently  be  impossible  to  draw  a  tangent  to  the  hyper 


INDETERMINATE    GEOMETRY. 


163 


bola  parallel  to  a  given  line,  when  the  diameter  to  be  drawn  par- 
allel to  the  second  chord,  does  not  intersect  the  curve. 


^^ 


139.     If  in  the  equation 


aYy"   +  h^-e'x"   =  a^'i. 


.(1), 


Art.  (133),  x"  and  y"  be  regarded  as  variables,  it  will  be  the 
equation  of  a  right  line ;  and  since  both  values  of  x"  and  y"  de- 
duced from  equations  (2)  and  (3),  Art.  (133),  must  satisfy  this 
equation,  the  right  line  must  pass  through  both  points  of  contact, 
or  will  be  the  indefinite  chord  which  joins  them^ 

If  any  point,  as  O,  be  taken  upon  this  chord,  its  co-ordinates, 
which  we  denote  by  c  and  d,  will  satisfy  equation  (1),  and  give  the 
condition 

a^d  +   b^x'c  =  a%^ (2). 


Every  set  of  values  for  x'  and  y'  which  will  satisfr  this 
equation,  will  give  a  point 
from  which,  if  two  tangents 
be  drawn  to  the  eUipse,  the 
chords  joining  the  points  of 
contact  will  pass  through  the 
^  point  O^  Hence,  if  y'  and  x' 
be  regarded  as  variables  in 
this  equation,  it  will  represent  a  right  line,  every  point  of  which 
will  fulfil  the  above  condition. 

As  in  Art.  (95),  this  line  is  the  polar  line  of  the  pole  0. 

If  through  the  point  0  and  the  centre,  a  right  line  be  drawn, 


its  equation  will  be 


X, 


:J^ 


1€^  INDETERMINATE    GEOMETRl. 

If  this  equation  be  combined  with  the  equation  of  the  ellipwi^ 
(e),  Art  (105),  we  find  for  the  co-ordinates  of  the  point  M, 

abc  ahd 


Substituting  these  for  x"  and  y''  in  the  equation  of  the  tangent 
line,  (3),  Art.  (128),  we  have  for  the  equation  of  the  tangent  at 
the  point  M, 

a»c?y  +  b^cx  =  ab  Va^d^  -f-  b^c^, 

which  is  evidently  parallel  to  the  polar  line,  represented  by  eqna* 
tion  (2). 

If  the  line  OC  be  produced  until  it  intersects  the  polar  line  NN' 
in  N  ;  for  this  point  we  shall  have 


^  —    ^  and  ***^'  —  — 

"/   ■""  T  a^'  ~~  M 


hence,  the  chord  which  joins  the  points  of  contact,  M'  and  B, 
of  two  tangents  drawn  from  N,  in  this  case  represented  by  equa- 
tion (1),  will  also  be  parallel  to  the  polar  line. 

These  properties  give  the  following  constructions. 

Mrst.  The  pole  being  given,  to  construct  the  corresponding 
polar  line. 

Through  the  pole  and  centre,  draw  the  line  (XJ ;  at  the  point 
M,  in  which  it  intersects  the  curve,  draw  a  tangent ;  through  the 
pole  draw  a  chord  parallel  to  this  tangent ;  at  either  point,  as  M', 
in  which  this  chord  intersects  the  curve,  draw  a  second  tangent ; 
through  the  point  N,  in  which  this  intersects  the  Hne  CO  produced, 
draw  a  line  parallel  to  the  first  tangent,  it  will  be  the  required 
polar  line. 

Second.  The  polar  line  being  given,  to  construct  the  correspond- 
ing pole. 

Draw  a  tangent  parallel  to  the  polar  line ;  join  the  point  of  con- 
tact M,  with  the  centre,  and  produce  this  line  until  it  meets  the 


INDETERMINATE    GEOMETRY.      '  ICfi 

polar  line  in  N ;  through  this  point  draw  a  second  tangent  NM', 
and  through  the  point  of  contact,  M',  draw  a  choriii.  M'O,  parallel 
to  the  polar  line ;  the  point  in  which  it  intersects  the  line  MC  will 
be  the  pole. 

It  should  be  remarked,  that  if  the  given  line  cuts  the  ellipse, 
this  construction  will  fail,  as  the  point  N  will  lie  within 
the  ellipse  and  no  tangent  can  be  drawn  from  it. 

When  the  ellipse  becomes  a  circle,  the  line  CM  becomes  per 
pendicular  to  the  tangent  at  M  and  also  to  the  polar  line,  and  tlic 
above  constructions  are  much  simplified. 
Thus,  to  construct  the  polar  line  : 
Through  the  given  pole  draw  a  line  to 
the  centre  ;  draw  a  second  line  perpen- 
dicular to  this,  at  the  pole ;  at  either 
point  in  which  this  perpendicular  inter- 
sects the  circle  draw  a  tangent ;  through 
the  point  N,  in  which  this  tangent  intersects  the  line  drawn  to  the 
centre,  draw  a  line  perpendicular  to  the  last  line ;  it  will  be  the 
polar  line. 

To  construct  the  pole :  Through  the  centre  draw  a  line  perpen- 
dicular to  the  polar  line ;  from  the  point  in  which  it  intersects  it, 
draw  a  tangent ;  from  the  point  of  contact  draw  a  perpendicular  to 
the  first  line  ;  the  point  in  which  it  intersects  it  will  be  the  pole. 

140.  The  equations  of  the  preceding  article  become  the  corres- 
ponding equations  of  the  hyperbola,  by  changing  h^  into  —  h\ 
and  it  will  be  readily  seen  that  the  properties  of  the  polar  line  and 
the  constructions  are  precisely  the  same  as  for  the  ellipse. 

When  it  is  impossible  to  draw  a  tangent  to  the  hyperbola  par- 
allel to  a  given  line.  Art.  (138),  the  construction  will  fail. 


141.     The  equation  of  any  straight  line  passing  through  the 
point  of  contact  of  a  tangent  to  an  ellipse,  will  be  of  the  form 


166 


INDETERMINATE    GEOMETRY, 
y    -    y/    =    d\x    -    X") (1). 


If  this  line  is  perpendicular  to  the  tangent,  we  must  have,  Ai-t. 
(28), 

dd'  -{-  \  ^  0,  or  c?'  =   —  J-. 


But,  Art.  (128), 


whence 


d  '-=   - 


h*x^ 


and  equation  (l)  beconiies 


aV" 


y  -y"  -  wtM  -  ^"> (2), 


5V 


for  the  equation  of  a  normal  to  the  ellipse^  x\rt.  (98). 
K  we  make     y  =  0,     in  equation  (2),  we  deduce 


L 


X"    -^    X 


h^x" 


C     R    P      B 


in  which  x  is  the  distance  CR, 
and 


or 


X"  —  a;  =  CP  —  CR  =  RP  =    thesuhnormal. 
If    a  =i  b,     equation  (2)  becomes 

y  -  y"  =  yl(x  -  ar") 
x" 

yx"   —  2/";c  r=  0. 


%jr^ 


>^  V 


1^  r  * 


W. 


INDETERMINATE    GEOMETRY 


16' 


As  there  is  no  absolute  term  to  this  equation,  the  normal  to  the 
circle  passes  through  the  centre,  Art.  (38). 


142.  On  the  transverse  axis  of  the  ellipse  let  a  semi-circle  be 
described,  and  within  this  semi-circle  let  us  inscribe  any  pqlygon, 
AN'NB.     From  the  vertices  of  this  poly-  J^, 

gon  draw  ordinates  to  the  transverse  axis, 
and  join  the  points  in  which  they  inter- 
sect the  ellipse,  thus  forming  a  polygon 
AM'MB,  of  the  same  number  of  sides. 

if  the  ordinates  of  the  points  N,  N',  <fec.,  be  denoted  by  Y,  Y', 
&c.,  and  the  corresponding  ordinates  of  M,  M',  by  y,  y',  &c.,  the 
abscissas  being  x,  a;',  <fec.,  we  shall  have,  Art.  (12*7), 


C    I*      Pr   n 


whence 


Y   :   y   :  :  a   :   J,  Y'   :   y'   :  :   a   :   5 ; 


Y  +  Y'    :   y  -f  y'   :  :   a   :   6. 


The  area  of  the  trapezoid  PNN'P',  forming  a  part  of  the  poly* 


ffon  in  the  circle,  will  be 


odl 


and  the  area  of  the  corresponding  trapezoid,  PMM'P', 

These  expressions   being  equi- multiples   of     Y  +  Y'      and 
y  +  y',     are  to  each  other  as 

Y  4-  Y'   :   y  +  y',        or  as        a   i   h. 

In  the  same  way,  it  may  be  proved  that  any  trapezoid  in  the 


168  INDETERMINATE    GEOMETRY. 

circle  is  to  the  corresponding  one  in  the  ellipse  as  a  is  to  5  ;  hence, 
the  sum  of  all  in  the  circle,  or  the  polygon,  AN'NB,  will  be  to  the 
sum  of  all  in  the  eUipse,  or  the  corresponding  polygon  AM 'MB,  as  a 
is  to  5  ;  and  this  will  be  true,  whatever  be  the  number  of  the  sides. 
If  now  the  number  of  sides  be  indefinitely  increased,  the  areas 
of  the  polygons  will  become  equal  to  the  areas  of  the  circle  and 
ellipse  respectively,  and  we  shall  have  the  first  is  to  the  second  as  a 
is  to  6  ;  or  denoting  the  area  of  the  circle  by  S,  and  that  of  the 
elHpse  by  s,  we  shall  have 

S   :   s  :  :   fl   :  6 ;  whence  s  =  —  S, 


a 


and  substituting  for  S  its  value  -n-a^, 

s  =  <aft, 

or  the  area  of  an  ellipse  is  equal  to  the  rectangle  upon  its  semi-axes 
multiplied  hy  the  ratio  of  the  diameter  to  the  circuviference  of  a 
circle. 

The  above  expression  may  be  put  under  the  form 


that  is,  the  area  of  the  ellipse  is  a  mean  proportional  between  the 
areas  of  the  two  circles  described,  one  upon  the  transverse,  and  the 
other  upon  the  conjugate  axis. 


OP     CONJUGATE     DIAMETERS     OP     THE     ELLIPSE     AND 
HYPERBOLA. 

143.  Let  it  now  be  proposed  to  ascertain  if  there  are  any  other 
co-ordinate  axes,  having  their  origin  at  the  centre,  to  which,  if  the 
ellipse  and  hyperbola  be  referred,  their  equations  will  have  the 
same  form  as  when  referred  to  their  centres  and  axes,  Arts.  (106) 


INDETERMINATE    GEOMETRY.  lOd 

and  (lOY).  For  this  purpose  let  us  take  formulas  (3),  Art.  (67), 
and  substitute  the  values  of  x  and  y  in  equation  {e)^  we  thus  obtain 

a\x"^  sin2  a  +   2a;y  sin  a  sin  a'   +  y  2  sin^  a') 

+  h\x'^  cos'^  a  +   2a;'y'  cos  a  cos  a!   +  y'^  cos*  a')   =  a*6«, 

or  arranging  and  omitting  the  dashes  of  the  variables, 

(ov^sin^a'   +  S*  cos*  a  V  +   (a=^  sin^  a   +   6*  cos*  a>« 

4-  2(c2  sin  a  sin  a'   +  ^*  cos  a  cos  aJ^xy  =  a%^ .(1), 

which  is  the  equation  of  the  ellipse  referred  to  any  set  of  oblique 
axes,  having  the  origin  at  the  centre.  This  equation  will  be  of  the 
same  form*  as  equation  (e),  if  the  term  containing  xy  be  made  to 
disappear,  which  requires  that 

a*  sin  a  sin  aJ  +  6'  cos  a  cos  otf   =   0 (2). 

llie  substitution  of  this  condition  in  equation  (1),  reduces  it  to 

^a'sin*  a'  +  6*cos«a')y2  +  (a*  sin*  a  +  6*  cos*  a)x^  =  a*Z»*...(3). 

Making  y  and  x,  in  succession,  each  equal  to  0,  we  find 


v 


"■'•  CB', 


a*  sin*  a    +  6*  cos*  a 

,  =  ±  J      "'*'     :  =  CD', 

^  a*  sin*  a'  +  i*  cos*  a' 

both  of  which  values  are  real  for  all  values  of  a  and  u' ;  hence,  the 
curve  cuts  each  axis  of  co-ordinates 
in  two  points,  on  different  sides   of 
the  centre,  and  at  equal  distances. 

If  we   place  these   distances  re-      v:^^ 
spectively   equal   to  a'   and   b',  we 
have 


170 


r'«    — 


INDETERMINATE    GEOMETRY. 


a^  sin-  a  4-  ^'^  cos^  a 
from  which 


J/2    ^ 


a*  sin*  CL  -\-h^  cos^  a 


a2ja 


a^  sin2  a'  +  h^  cos*  a' 


a«  sin*  a.'  +  i*  cos*  a'   =  —  , 


Substituting  these  values  for  the  coefficients  of  x^  and  y^  in 
equation  (3),  and  striking  out  the  common  factor  a*6*,  we  have 


6'*   ^ 


=   1, 


or         a'V   +  ^^'^^  =  a'*^"* (e% 


an  equation  of  precisely  the  same  form  as  equation  (e),  and  which 
if  solved  as  in  Art.  (106),  will  give  for  each  value  of  a;  <  a\ 
two  values  of  y  equal  with  contrary  signs,  and  these  taken  to- 
gether will  form  a  chord  mm',  which  is  bisected  by  the  axis  of  X  ; 
hence,  this  axis  is  a  diameter  of  the 
ellipse.  Art.  (100).  By  solving  equa- 
tion (e')  with  reference  to  ar,  it  may 
also  be  proved  that  the  axis  of  Y  is  a 
diameter  and  bisects  a  system  of  chords 
parallel  to  the  axis  of  X.  These  di- 
ameters are  called  conjugate  diameters; 
and  in  general,  two  diameters  are  conjugate^  when  each  bisects  a 
system  of  chords  parallel  to  the  other. 

If  in  equation  (e')  we  make     a;  =   d:  a',     we  deduce 


±  0; 


hence,  the  ordinates  at  A'  and  B',  produced,  are  tangent  to  the 
curve,  Art.  (34). 
li    y  =   zt  5', 


.u  =   =t  0. 


INDETERMINATE    GEOMETRY.  l7l 

Hence,  the  tangent,  at  the  vertex  of  either  diameter,  is  parallel 
to  its  conjugate,  or,  to  the  chords  which  the  diameter  bisects. 

Equation  [e')  is  called  the  equation  of  the  ellipse  referred  to  its 
centre  and  conjugate  diameters,  in  which  a'  and  6'  are  the  semi- 
conjugate  diameters. 

I      144.     Since,  whenever  a  and  a'  have  such  values  as  to  satisfy 

^.-aquation  (2^  of  the  preceding  article,  the  axes  of  co-ordinates  be* 

,       come  conjugate  diameters,  that  equation  is  called,  the  equation  of 

I       condition  for  conjugate  diameters,  in  which  a  and  a'  are  the  angles 

r        formed  by  these  diameters  respectively,  with  the  transverse  axis. 

V-.  Dividing  by  cos  a  cos  a',  and  recollecting  that 


/ 


sm  a         ,  sin  a'         ,  . 

=  tang  a,  =  tang  a', 

cos  a  cos  a' 


wo  may  put  the  equation  under  the  form 

tang  a  tang  a'  =   —  — (1). 

Since  a  and  a'  are  indeterminate  in  this  equation,  it  follows  that 
there  is  an  infinite  number  of  conjugate  diameters,  and  if  a  partic- 
ular value  be  assigned  to  a  or  a'  the  corresponding  value  of  the 
other  will  be  determined  and  the  position  of  the  diameters  known. 

If    a  =  0,     tang  a  =  0,     and  equation  (1)  gives 

tang  a'   =    00 ,  a'   =   90°. 

If    a'  =  0,     tang  a'  =  0,     whence 

tang  a  =   00  ,  a  =   90°. 

Hence,  if  either  diameter  coincides  with  the  transverse  axis  the 
other  will  coincide  with  the  conjugate.  Also,  if  either  a  or  a'  is 
90°  the  other  will  be  0 ;  that  is,  if  either  diameter  coincides  with 
the  conjugate  axis,  the  other  will  coincide  with  the  transverse ; 
and  the  axes  are  conjugate  diameters. 


l72  INDETERMINATE    GEOMETRY. 

145.     If  any  conjugate  diameters,  except  the  axes,  are  at  right 
angles,  we  must  have,  Art.  (28), 

tang  a  tang  a'  =  —  1  ; 
flko  (Art.  144), 


tang  a  tang  a    = j. 


both  of  which  cannot  be  satisfied  by  any  vahies  of  a  and  a',  ex- 
cept a  =  0,  and  a'  =  90°,  or  a  =  90^,  and  a'  =  0 ;  in 
which  case,  as  seen  above,  the  diameters  coincide  with  the  axes: 
hence,  the  axes  are  the  onbj  conjugate  diameters  at  right  angles. 

Vi  a  ■=.  b^  both  equations  become  the  same,  and  may  be  satis- 
fied by  any  value  of  a  with  the  corresponding  deduced  vahie  of 
a' ;  hence,  in  a  circle,  any  two  conjugate  diameters  are  at  right 
angles. 


146.     By  comparing  equation  (l).  Art.  (144),  with  equation 
(1),  Art.  (135),  we  see  that 

cc'  =  tans:  a  tanoj  a' ; 

hence,  if    c  =  tang  a, 

c'  =  tang  a', 

and  the  reverse ;    that  is,  if  one  of  tico  s^ipplementary  chords  is 
parallel  to  a  diameter,  the  other  will  b"  parallel  to  its  conjugate. 


147.     If  in  equation  (1)  of  Art  (144),  we  put   —  6«  for  ^, 
we  iiave 


INDETERMINATE    GEOMETRY, 


173 


tang  a  tang  a'  = 


a" 


(1). 


which  is  ike  equation  of  condition  for  conjugate  diameters  in  ike 
hyperbola^  and  admits  of  the  same  discussion,  and  gives  precisely 
the  same  results  for  the  hyperbola,  as  were  deduced  above  for  the 
eUipse. 

If    a  =  b,     we  have 


tanjx  a 


tanof  a' 


=  cot  a' 


.  hence,  in  the  equilateral  hyperbola,  the  conjugate  diameters  fonn 

^  angles  with  the  transverse  axis,  which  are  complements  of  each 
other. 


r 148.     If  in  equation  (3),  Art.  (143),  we  put    —  b^  for  6«,    it 

becomes 

(a^sin^a'  —  b^cos^a')y^  +  (a«sin»a  —  b^cos^a)x^  =  —  a^b^...{l), 
and  making  y  and  »,  in  succession,  each  equal  to  0,  we  find 


J n^!^! 

^  a»  sin«  a  —   6*  cos'*  a 


^■'-^Vr, 


-   aVj» 


a*  sin**  a'  —   6*  cos'  a' 


The  reality  of  these  values  will  depend  upon  the  sign  of  the  de- 
nominator under  the  radical  sign.  K  that  of  the  first  is  negative, 
X  will  be  real.     In  this  case 


^:  a*  sin«  a  —   6«  cos«  a   <   0, 


cos'  a 


<  — »      tang  a  <  -  ; 

n*  rr 


hence,  from  equation  (1)  of  the  preceding  article,  we  have 


174 


INDETERMINATE    GEOMETRY. 


^       tang  a'   > 


cos*  a 


;  > 


6* 


a*  sin*  aJ  —  IP-  cos*  a'  >  0, 


and  the  denominator,  under  the  second  radical  sign,  is  positive, 
and  the  value  of  y  imaginary. 

In  the  same  way,  it  may  be  shown  that  if  y  is  real,  x  must  be 
imaginary.  Therefore,  if  one  of  the  conjugate  diameters  of  the 
hyperbola  cuts  the  curve  the  other  will  not,  and  the  converse. 

If  then,  we  regard  the  above  value  of  x  as  real,  we  may  place  it 
equal  to  a',  and  the  imaginary  value  of  y  equal  to  6'  V  •—  1, 
whence 


r/«    — 


-    rt*6* 


a'  sin*-a  —   IP  cos*  a 


i'* 


--  a*6* 

a* sin*  a'  —  6*  cos*  a'' 


from  which,  deducing  the  values  of  the  denominators,  and  substi- 
tuting in  equation  (1),  we  have 


?^  -  ^  =   -   1 

6'*         a'* 


or      a'*i/*  —  6'*a;*  =   —  a'*6'2 {h% 


for  the  equation  of  the  hyperbola  referred  to  its  centre  and  conju- 
gate diameters,  in  which,  a'  and  V  are  the  semi-conjugate  diame- 
ters. 

This  equation  is  of  the  same  form  as  equation  (A),  Art.  (107), 

and  from  it  we  may  prove 
as  in  Art.  (143),  that  each 
diameter  bisects  a  system  of 
chords  parallel  to  its  conju- 
gate, or  parallel  to  the  tan- 
gent at  its  vertices,  if  it  have 
vertices.  If  a  second  hy- 
perbola be  described  upon 
DE  as  a  transverse  axis, 
having  BA  for  its  conjugate,  it  is  said  to  be  conjugate  to  the  first 


INDETERMINATE    GEOMETRT.  Il5 

hyperbola ;  that  is,  two  hyperholas  are  conjugate  when  the  transverse 
axis  of  one  is  the  conjugate  of  the  other,  and  the  reverse. 

Tlie  equation  of  the  conjugate  hyperbola,  obtained  by  changing 
X  into  y,  Art.  (108),  and  a  into  b,  in  equation  (A),  is 


a-g 


2  -   h^x^  =  a^b^. 


K 


149.  The  parameter  of  any  diameter  of  either  tiie  ellipse  or 
hyperbola,  is  a  third  proportional  to  the  diameter  and  its  conju- 
gate, the  conjugate  being  the  mean.  Thus,  for  the  parameter  of 
2a'  =  B'A' 

o7/8 

2a'    :    2b'    :  :    2b'    :    2p ;       whence         2p  =.- 

a' 

2&* 
The  parameter  of  the  transverse  axis,     —  ,     is  also  the  para- 

a 

meter  of  the  curve,  Art.  (115). 

For  the  parameter  of  the  conjugate  axis,  we  have 

2a« 
^  b 


150.  As  equations  {e')  and  (^'),  Arts.  (143),  (148),  are  pre- 
cisely the  same  as  equations  {e)  and  (A),  except  that  a'  and  b'  en- 
ter instead  of  a  and  6,  it  follows  that  any  algebraic  expression  de- 
duced from  the  latter,  will  become  the  corresponding  expression 
for  the  former,  by  changing  a  into  a'  and  b  into  b'.  Thus  the  pro- 
portions of  Arts.  (125),  (126),  become 

y'«    :    y"^    :  :    {a'  +  x'){a'   —  x')    :    (a'   +  x"){a>  —  x"), 
y'*    :    y"*    :  :    (x'  +  a')(x'   -  a')    :    (x"  +  a'){x''  -  a')  \ 

the  first  of  which  shows  that,  the  squares  of  the  ordinates  drawn  tj 
mny  diameter  of  an  ellipse,  are  to  each  other  ««  the  rectangle  of  the 


176  INDETERMINATE    GEOMETRY. 

segments  into  which  the  diameter  is  divided  ;  and  tlie  second  that, 
the  squares  of  the  ordinates  drawn  to  any  diameter  of  an  hyperbola, 
which  intersects  the  curve,  are  to  each  other  as  the  rectangles  of  the 
distances  from  the  foot  of  each  ordinate,  to  the  vertices  of  the  di- 
ameter. 

These  properties  enable  us  to  construct  either  curve,  having 
given  tvro  conjugate  diameters  and  the 
angle  formed  by  them.  Thus,  let  A'B' 
and  D'E'  be  two  such  diameters.  Re- 
volve D'E'  until  it  becomes  perpen- 
dicular to  A'B' ;  on  the  two  as  axes, 
describe  an  ellipse  (or  hyperbola),  in 
which  draw  any  number  of  ordinates  mp,  ni'p',  &c. ;  then  revolve 
these  until  they  become  parallel  to  the  primitive  position  of  D'E', 
their  extremities  will  be  points  of  the  curve. 


151.    The  equations  of  the  tangent.  Arts.  (128),  (131),  become, 
•when  referred  to  conjugate  diameters, 

a"»yy"   -  h'Hx"  =    -  a'«6'«, 
the  first  for  the  ellipse,  and  the  second  for  the  hyperbola. 


152.  The  equations  of  condition  for  supplementary  chords, 
Arts.  (135),  (136),  when  drawn  from  the  extremities  of  a  diameter 
2a',  become 

cc'  =  —  — ,    for  the  ellipse (1), 

a'* 

cc'  =  —  ,    for  the  hyperbola, 
a'* 

in  which,  since  the  axes  of  co-ordinates  are  oblique,  c  and  c'  r© 


INDETERMINATE    GEOMETRY. 


17V 


present  tlie  ratio  of  the  sines  of  the  angles  which  the  chords  make 
with  the  axes,  Art.  (20). 


153.  Likewise,  equation  (1),  Art.  (13V),  will  belong  to  a  di- 
ameter and  tangent  at  its  vertex,  when  referred  to  conjugate  di- 
ameters, if  w^e  change  a  into  a'  and  I  and  Z>',  and  regard  d  and  d' 
as  the  ratio  of  the  sines,  &c. ;  thus. 


dd'  = 


h"' 


Comparing  this  with  equation  (1)  of  the  preceding  article, 
have 


we 


cc' 


dd' 


and  the  same  for  the  hyperbola.  Hence,  if  c  =  c?,  c'  =  d' 
and  the  converse.  Therefore,  in  either  curve,  if  a  chord,  drawn 
from  the  extremity  of  a  diameter,  is  parallel  to  a  tangent,  its  sup- 
plement will  be  parallel  to  the  diameter  passing  through  the  point, 
of  contact,  and  the  converse.  Also,  if  one  of  two  supplementary 
chords  is  parallel  to  a  diameter,  the  other  will  be  parallel  to  its 
conjugate :  or,  a  set  of  supplementary  chords  may  always  be  drawn 
from  the  extremities  of  any  diameter  parallel  to  a  set  of  conjugate 
diaftieters. 


154.  The  properties,  of  supplementary  chords,  diameters,  and 
tnngents,  discussed  in  the  preceding  article,  give  the  following 
constructions. 

First   If  either  the  ellipse  or  hyperbola  is  traced  upon  paper ; 

draw  any  two  parallel  chords,  nn  and  ^ _^ 

n'n',  and    bisect   them   by    a   straight 

line,  this  will  be  a  diameter.  Art.  (100). 

If  two  diameters  be  thus  constructed, 

their  intersection  will  be  the  centre  of 

the  curve. 

12 


Jd 

^^^ 

V 

\^Vb- 

><^. 

^ 

<^ 

c 

y 

178 


IN'DHTERMIXATE    GEOMETRY. 


Second.  On  any  diameter,  A'B',  found  as  above,  describe  a 
semi-circle  and  draw  two  chords  from  the  point  M,  in  which  it  in- 
tersects the  curve  to  the  extremities  of  the  diameter ;  these  chords 
will  be  supplementary  and  perpendicular  to  each  other  ;  draw  two 
diameters  parallel  to  these  and  they  will  be  the  axes. 

And,  in  general,  to  construct  a  set  of  conjugate  diameters 
making  a  given  angle  with  each  other.  Upon  any  diameter  de- 
scribe an  arc  capable  of  containing  the  given  angle ;  from  the 
point  in  which  it  cuts  the  curve,  draw  two  chords  to  the  extremi- 
ties of  the  diameter  ;  through  the  centre  draw  two  diameters  par- 
allel to  these  chords ;  they  will  be  the  required  diameters. 

Third.   If  one  diameter,  as  D'E'',  is  given,  and  it  be  required  to 

construct  its  conjugate. 
From  the  extremity  of  any 
diameter  draw  a  chord  Rm, 
parallel  to  the  given  di- 
ameter; draw  the  supple- 
ment, Om,  of  this  chord ; 
through  the  centre  draw  a 
diameter  CA'  parallel  to 
this  supplement,  it  will  be 
the  one  required. 

Fourth.  To  draw  a  tangent  at  a  given  point,  as  A'.  Join  this 
point  with  the  centre  ;  through  the  extremity  0,  of  any  diameter, 
draw  a  chord  parallel  to  CA' ;  draw  the  supplement  of  this  chord, 
Rm  ;  parallel  to  which,  draw  a  line  through  the  given  point ;  il 
will  be  the  required  tangent. 

Fifth.  To  draw  a  tangent  pnrallo]  to  a  given  line,  as  L.  Make 
the  same  construction  as  in  Art.  (137),  using  any  diameter  as  OR, 
instead  of  the  transverse  axis.  Or  thus :  draw  any  two  chords 
mR,  7/i'R',  parallel  to  the  given  Hne ;  bisect  them  by  a  straight 
line ;  the  points  A'  and  B',  in  which  this  intersects  the  curve  will 
be  the  points  of  contact,  through  either  of  which  draw  a  line  par 
allel  to  the  given  line,  it  will  be  the  required  tangent. 


/ 


INDETERMINATE    GEOMETRY.  l70 

155.     Let  CB'  and  CD'  be  any  two  semi-conjugate  diameters 
of  the  ellipse.     On  the  transverse  axis  AB, 
describe  a  semi-circle ;  through  the  points        i»^ 
B'  and  D'  draw  two  ordinates  and  produce       /^"^ 


^ 


ihem  to  M  and  M' ;  draw  the  radii  CM 
and  CM',  and  denote  the  angles  MCB  "^  ^'  ^  ^  ^ 
and  M'CB  by  /3  and  /3'.  The  right  angled  triangles  CPB'  and 
0PM,  give 

^  tang  a    :    tang  /3    : :    PB'    :    PM    \\    h    \    a, 

Art.  (127) ;  hence 

tang  a  =  _  tang  /3. 
a 

Also,  the  triangles  OP'D'  and  OP'M',  since  the  angles  at  the 
I         bases  are  supplements  of  a'  and  /3',  give 

tang  oJ  zz=  -  tang  /3'. 
a 

Multiplying  these  equations,  member  by  member,  we  have 

tang  a  tang  a'   =r  _  tang  /3  tang  /3'  =   —  —  , 
a*       '  a« 

Art.  (144) ;  hence 

tang  /3  tang  /8'  =   —   1, 

and  the  two  radii  CM  and  CM'  are  perpendicular  to  each  other. 
Art.  (28) ;  therefore 

Q'  =  90°  -f  /3,  sin  /3'  =  cos  /?,         cos  /3'  =   -  sin  /?• 

150.     From  the  triangles  CPB'  and  CPM,  we  ha\e 


p'  c    f 

180  INDETERMINATE    GEOMETRY. 

CP  =  a'  COS  a,  CP  =  a  cos  /?, 

PB'  =  a'  sin  a,  PM  =  a  sin  jS ; 

whence,  from  the  first  two, 

a'  COS  a  =  a  cos  ^ (1), 

and  from  the  second 

al  sin  a    :    a  sin  ^    : :    PB'    :    PM    :  :    6   :    ft^ 

a*  sin  a  =   h  sin  /3 (2). 

In  the  same  way,  the  triangles  CP'D'  and  CP'M'  give 

b' cos  a'  =  a  cos  (3'  =   —  asin^ (3), 

h'  sin  a'   —    h  sin  /3'  =  6  cos  ^ (4), 

after  substituting  for  cos  /3'  and  sin  ^'  their  vakies,  as  deduced  in 
the  preceding  article. 

Multiplying  equations  (1)  and  (4),  member  by  member,  and 
then,  (2)  and  (3),  and  subtracting  the  latter  product  from  the 
former,  we  have 

«'6'(sin  a'  cos  a  —  sin  a  cos  a')  =   ah(cos^  /3  -f  ^i-*  A^  f 

or 

a'h'  sin(a'  ~  a)  =  ah (o). 

Squaring  both  members  of  (1)  and  (3)  and  adding,  we  have 

a'2cos«a  +  7/*cos«a'  =  a» 
tn  the  same  way,  from  (2)  and  (4)  we  obtain 

a'2  sin^  a   -f  6'^  sin^  a'  =  6^. 
Adding  the  last  two  equations,  member  by  member, 
a'2  +  h'^  =  a«  4-  ^ (6). 


INDETERMINATE    GEOMETRY. 


181 


157.     If  we  unite  equation  (1)  of  Art.  (144),  with  (5)  and  (6) 
of  the  preceding  article,  we  have 


tang  a  tang  a'  =   —  — 

a'h'  sin  (a'   —   a)   =  a5 
a'a  +  j/2  ==  ^2   .|.   2,2. 


.(1), 
•(2), 


three  equations  containing  six  quantities,  either  three  of  which  be- 
ing given,  the  others  may  be  determined. 

If  the  angle     ol'  —   a,     made  by  the  conjugate  diameters  with 
each  other,  is  given  equal  to  w,  we  have 


a'    —   a  =   w. 


tang  a'  =   tang  (w   +  «)» 


and  this  value  in  equation  (1),  will  give  an  equation  from  which 
tang  a  may  be  found,  and  thus  both  a  and  a',  become  known. 
Let  us  resume  equation  (2), 


a'b'  sin  (a'   —    a)   =  ah. 


•(2), 


and  at  the  vertices  of  any  two  conjugate  diameters  A'B'  and  D'E', 
draw  tangents  forming  a  parallelo- 
gram. Also  at  the  vertices  of  the 
axes,  draw  tangents  forming  a  rect- 
angle. From  the  right  angled  tri- 
angle D'RC,  we  have 

D'R  =  D'CsinD'CR  =ft'sin(a'-a) ; 

hence,  the  first  member  of  equation  (2)  is  equal  to  CB'  x  D'R, 
or  the  area  of  the  parallelogram  CQ,  v^hile  the  second  member  is 
equal  to  CB  x  CD,  or  the  area  of  the  rectangle  CS.  If  these 
equals  be  multiplied  by  4,  we  have  four  times  the  parallelogram 
CQ,  or  the  parallelogram  QQ',  equal  to  four  times  the  rectangle 
CS,  or  the  rectangle  SS' ;    that  is,  the  parallelogram  constructed 


182 


INDETERMIKATE    GEO! 


itettV^'  * 


upon,  any  two  conjugate  diameters  qfihe  elltpsey  is  equivalent  to  thU 
rectangle  upon  the  axes. 

Multiplying  both  members  of  equation  (3),  by  4,  we  have 

4a^2  +  46'2  =  4a2  +  Ah\ 

But 

4a'«  =  (2a')S  46'^  =  (26')*  Ac; 

hence,  the  sum  of  the  squares  upon  any  two  conjugate  diameters  of 
the  ellipse^  is  equal  to  the  sum  of  the  squares  upon  the  axes. 


M 


158.  If  in  equations  (1),  (2)  and  (3),  of  the  preceding  article, 
we  change  6*  into  —  b^  and  b'^  into  ~  b'%  we  have,  for  the 
hjrperbola, 

■      ■     ■    ''       (1). 

(2), 

(3), 

from  which  either  three  of  the  quatitities  may  be  determined  when 
the  others  are  known. 

If  a'  —  a  =  w,  is  given,  a  and  a'  may  be  determined  as  in 
the  pi-eceding  article. 

The  perpendicular  D'R  is  equal  to 

D'C  sin  D'CR  =  b'  sin  (a'  -  a) ; 


,    .A,    1-^ 

*"fe 

a«" ^ 

a'b' 

sin 

K 

— 

«) 

= 

ab 

a"^ 

__ 

j'« 

=; 

a» 

b^ 

X 


y 


/ 


/^ 

^ 

^ 

M 

hence,  the  first  member  of 
equation  (2)  is  equal  to 
CA'  X  D'R,  or  the  area 
of  the  parallelogram  CQ  ; 
while  the  second  member 
is  equal  to  the  rectangle 
CS.  Multiplying  these 
equals  by  4,  we  obtain  the 


INDETpR^INAT|:    GEOMETRY. 

parallelogram  amstructed  upon  any  two  conjugate  diameters  of  iha 
Injperlola  equivalent  to  the  rectangle  upon  the  axes. 
From  equation  (3),  vre  have 

4a'2  __   4^/2  =  4a2  —   45«, 

that  is,  /Ae  difference  of  the  squares  of  any  two  conjugate  diameters 
of  the  hyperbola^  is  equal  to  the  difference  of  the  squares  of  the  axes. 


159.  If  the  conjugate  diameters  of  an  elHpse  are  equal  to  each 
other,  the  two  expressions  for  a'^  and  6'^,  Art.  (143),  must  be 
equal,  which  requires  that 

a^  sin*  a  -{-  i*  cos*  a  =  a*  sin*  a'   -{-  6*  cos*  a', 

and  every  set  of  values  of  a  and  a'  which  will  satisfy  this  equation, 
provided  they  at  the  same  time  satisfy  equation  (1),  Art.  (144),  will 
give  the  position  of  a  set  of  equal  conjugate  diameters. 
Substituting  in  the  above  equjition 

sin*  a  =    1   —  cos*  a,  sin*  a'  =   1    —  cos*  a', 

it  becomes 

(a*  -   ?»*)cos*a  =   (a*  —  5*)  cos*  a' (1), 

which,  unless     a  =  b,     can  only  be  satisfied  by  making 

cos*  a  =  cos*  a',  or  cos  a   =    dc  cos  a'. 

The  first  value  cos  a  =  cos  a'  gives  a  =  a' ;  hence  the 
two  diameters  coincide  and  are  not  conjugate. 

The  second  value  cos  a  =  —  cos  a',  satisfies  equation  (1), 
Art.  (144),  and  requires  the  angles  to  be  supplements  of  each 
other,  or 

a'  +  a  r=   180°; 

hence,  Art.  (135),  the  diameters  must  be  parallel  to  the  supple- 
mentary chords  drawn  from  the  extremities  of  the  transverse  to 


184  INDETERMINATE    GEOMETRY. 

the  extremity  of  the  conjugate  axis.  They  will  therefore  make  a 
gi-eater  angle  with  each  other  than  any  other  set  of  conjugate 
diameters. 

If  a  =  b,  equation  (1),  will  be  satisfied  for  every  set  of 
values  of  a  and  a' ;  hence,  in  the  circle,  the  conjugate  diameters 
are  equal  to  each  other. 

When     a'  =   6',     equation  [e').  Art.  (143),  becomes 

a'V   +  a'^c^  =  a'\         or         y^  +  x^  =  a'\ 
for  the  ellipse  referred  to  its  equal  conjugate  diameters. 


160.  By  an  examination  of  equation  (3),  Art.  (158),  it  will  be 
seen  that  a'  can  not  be  equal  to  6',  unless  a  =  6  ;  that  is,  in 
the  hyperbola,  there  are  no  equal  conjugate  diameters,  except 
when  the  hyperbola  is  equilateral,  in  which  case  each  diameter  is 
equal  to  its  conjugate. 


OF    THE    HYPERBOLA    REFERRED   TO    ITS    ASYMPTOTES. 

t 

161.     If  in  equation  (1)  of  Art.  (143),  Ave  put    -   i^  for  ja^  it 
becomes 

(a'sin^a'   —   ft^cos^a^y*  +   (a^sin^a  —   V^  q,o'&^  a)x^ 

+  2(a'*sinasina'  —  ^''cosacosa'  )a;?/=  —  a^l)^ (1), 

for  the  equation  of  the  hyperbola  referred  to  any  set  of  oblique 
axes  having  their  origin  at  the  centre.  We  may  assign  such  values 
to  the  arbitrary  constants  a  and  a',  in  this  equation,  as  to  cause  the 
coefficients  of  y^  and  x^  to  be  0,  and  thus  have 

a«sin«a'  —   Z»^cos2a^=   0,         a^%\\\^cL  —  h^cos,^a  =  0 (2), 

whence  by  dividing  the  first  by     a"^  cos*^  a',     and  the  second  by 


INDETERMINATE    GEOMETRY. 


ISH 


a*  cos*  a,     aud  recollectino:  that 


sm' 


=  tang'^,     we  deduce 


tang  a'  =  ±  -, 


tang  a  =   ± 


But  it  is  evident  that  we  can  not  use  tang  a'  =  tang  a,  as 
in  such  case,  the  two  new  axes  of  co-ordinates  would  coincide.  lij 
therefore,  we  take 


we  must  take 


tang  a'  =  -  , 
a 


tang  a  =   — 


and  the  reverse. 

These  values  may  be  readily  constructed,  thus  :  At  the  vertei 
A,  erect  a  perpendicular 
to  BA,  on  which  lay  oft' 
the  distances  AL  and 
AL',  each  equal  to  h  and 
draw  the  lines  CL  and 
CL',  these  will  be  the 
new  axes  of  co-ordinates. 
For  the  right  angled  tri- 
angles CAL  and  CAL' 
give 

tang  ACL  =  tang  ACL'  =  _  =  tang  a'. 


But  the  tangent  of  ACL',  taken  with  a  contrary  sign,  is  equal  to 

the  tangent  of  360°  -  ACL',  and  also  equal  to  —-  =  tang  a; 

a 
hence 


186  ;  INDETERMINATE    GEOMBTBYldV 

360°  -   ACL'  =  a, 

and  CLMs  the  new  axis  of  X,  and  CL  the   new  axis  of  Y,  the 
angles  a  and  a'  being  as  marked  on  the  figure. 
Since 


CL  =   Va8 

+  K 

the  right  angled  triangle  ACL  gives 

sm  a'  —                         , 

cos  a'  = 

a 

Va«  +  J8 

Va«   +  6« 

and  since 

sin  a'  =   —  sin  a, 

cos  a'  - 

=  cos  a, 

we  also  have 

-  h 
sin  a  — 

cos  a  =  . 

a 

Substituting  these  values,  together  with  conditions  (2),  in  equa« 
lion  (1),  we  have 


{ —  I  ary  =  —  a' 


2  (  -^ :^  —  — ;; — ; — ^^  ]X7/  =  —  a^b^, 

whence 


a*  +  b^  ,ov 

a^y  =  — ^ — ,  or         xy  =  m,  (3), 

,    .            .  .     a*  +    b^ 
placing  m  for 

Solving  this  equation,  we  have 


m 

X 


INDETERMINATE    GEOMETRy. 


18^^ 


in  which,  as  a?  increases,  y  diminishes  ;  when  x  becomes  infinite  y 
becomes  0  ;  and  as  y  can  be  negative  for  no  positive  value  of  a:,  it 
follows  that  the  axis  of  X,  or  the  line  CL',  continually  approaches 
the  curve  and  touches  it  at  an  infinite  distance  without  cutting  it. 
By  solving  the  equation  with  reference  to  a?,  it  may  be  proved  that 
the  line  CL  enjoys,  the  same  property.  These  two  lines  are  called 
asymptotes  of  the  hyperbola ;  and  in  general,  an  asymptote  of  a 
curve  is  a  line^  which  continually  approaches  the  curve  and  becomes 
tangent  to  it  at  an  infinite  distance. 

By  an  inspection  of  the  figure,  it  is  readily  seen  that  the 
asymptotes  of  the  hyperbola  are  the  diagonals  of  the  rectangle 
described  on  the  axes. 

Equation  (3)  is  called  the  equation  of  the  hyperbola  referred  to 
its  centre  and  asymptotes. 

If  the  hyperbola  is  equilateral,  a'  =  45°,  tang  a'  =  1, 
the  angle  LCL'  =  90°,  and  the  asymptotes  are  perpendicular 
to  each  other. 


^ 


162.    If  we  take  the  expression.  Art.  (132), 


X  = 


CT, 


and  make    x"  =  a,    the  least  value  it  can  have  for  points  of  the 
curve.   Art.   (10*7),    wo 
shall  obtain 

CT  =  a  =  CA, 

which  is  the  greatest 
value  of  CT.  As  x"  in- 
creases, CT  diminishes 
until  x"  =z  CO,  when 
CT  =  0,  its  least  value, 
and  the  tangent  coincides  with  the  asymptote  ;  hence  all  tangents 


186 


INDETERMINATE    GEOMETRY. 


drawn  to  the  hyperbola  intersect  the  transverse  axis  between  the 
centre  and  the  vertex  of  that  branch  to  which  they  are  drawn  ; 
and  the  asymptotes  are  the  limits  of  all  tangents. 


^ 


163.  If  we  multiply  both  members  of  equation  (3),  Art.  (161), 
by  sin  2a',  the  sine  of  the  angle  LCL'  included  by  the  asymptotes, 
we  have 

xi/  sin  2a'  =  m  sin  2a'. 

The  second  member  of  this  equation  is  constant,  and  the  first 
for  any  point  of  the  curve,  as  M,  is 

CP  X  PMsinMPw    =  CP  X  Un, 

which  is  the  area  of  the  parallelogram  CPMR.     Hence,  the  areas 

of  all  parallelograms  de- 
scribed on  the  abscissas 
and  ordinates  of  points  of 
the  curve,  referred  to  the 
asymptotes,  are  equal,  each 
being  measured  by  the 
expression   m  sin  2a'. 

If  the  point  M  is  placed 
at  the  vertex  A,  the  par- 
allelogram becomes  the  rhombus  AOCO',  each  of  its  sides 
being 

-  CL  =  _  -v/a2   +   b\ 

Thk  rhombus,  described  on  the  abscissa  and  ordinate  of  the 
vertex,  is  called  the  power  of  the  hyperbola,  is  equivalent  to  the 
parallelogram  described  on  the  abscissa  and  ordinate  of  any  point 
of  the  curve,  and  as  is  readily  seen  from  the  figure,  is  one  eighth  of 
the  rectangle  described  on  the  axes. 


INDETERMINATE    GEOMETRY.  ,  189 

A 

In  the  equilateral  hyperbola 

2a'  =   90°,  sin  2a'  =   1, 

and  the  power  becomes  a  square,  the  area  of  which  is  m, 

164.     Let  x"  y"  denote  the  co-ordinates  of  any  point,  as  M, 
of  the   hyperbola.      The   equation   of  a 
right  line  passing  through  this  point,  will 
be  of  the  form 

y  -  y"  =  d{x  -  x«) (1). 

The  equation  of  the  hyperbola  being 

^y  —  ^n (2), 

the  condition  that  the  point  M  shall  be 
on  the  curve,  will  be 

x"y"  =  m. 


Subtracting  this  from  equation  (2),  we  have 


Combining  this  with  equation  (1),  by  substituting  the  value  of 
7/  taken  from  (1),  we  obtain 

i/"(x  -  x")  +  dx{x  —  X")  =  0,     or     {x  —  x"){y"  +  dx)  =  0. 

Placing  the  two  factors  of  the  last  equation,  separately,  equal  to 
0,  we  obtain  for  all  the  common  points,  Art.  (128,) 


X  ^  x"  =  0 

or 

=t  =  a:", 

y"   +  cZ:r  =   0 

or 

'  =  -  T ' 

(3). 


> 


100  INDETERMINATE    GEOMETRY. 

The  first  value  of  x  is  evidently  the  abscissa  of  the  point  M,  the 
second  must  then  be  that  of  M'. 

Now  if  the  line  MM'  be  revolved  about  M  until  the  point 
M'  coincides  with  M,  it  will  become  a  tangent,  and  the  value  of  x 
in  equation  (3)  will  become  x",  whence  we  have 

x" 
This  value,  in  equation  (1),  gives 

y  -   y"   =   _  ^  (^  -.   x'% 


for  tlie  equation  of  the  tangent^  referred  to  the  asymptotes. 
•    If  in  this  equation  we  make     y  =  0,     we  have 

^  X  —  x"  =  x"  =  PT, 

that  is,  the  suhtangent/is  equal  to  the  abscissa  of  the  j)oint  of  contact. 

And,  since  PT  =  CP,  we  have  MT  =  MS ;  that  is,  the 
part  of  the  tangent  included  between  the  asymptotes  is  bisected  at 
the  point  of  contact. 

If  in  equation  (1)  we  make     y  =  0,     we  find 

a;  _  a;"  =   -  ^  =   CT'  _  CP  =  PT', 
d 

the  same  value  found  in  equation  (3)  for  the  abscissa  of  M'  ;  hence 

A  PT'  =  MT', 

and  the  two  triangles  MPT'  and  M'P'T",  having  their  angles  also 
equal,  are  equal,  and  MT'  =  M'T"  ;  that  is,  if  any  straight 
line  be  drawn  cutting  the  hy2)erbola,  the  parts  included  between  the 
curve  and  asymptotes  will  be  equal. 

This  property  enables  us  to  construct  the  hyperbola  by  points 
A-hen  a  single  point  and  the  asymptotes  are  given.     Through  th« 


INDETETRMINATE '  GETOMETKY 


101      "» 


point,  as  M,  draw  any  right  line  ;  from  the  point  in  which  it  cuts 
one  of  the  asymptotes,  lay  off  the  distance  T"M',  equal  to  the  dis- 
tance MT',  from  the  given  point  to  that  in  which  it  cuts  tlie  other 
asymptote ;  the  extremity  of  this  distance  will  be  a  point  of  the 


curve. 


165.  If  a  tangent  TS  be  drawn  at  any  point,  M,  of  the  hyper 
bola,  and  the  half  tangent 
MT  be  denoted  by  /,  the 
half  diameter  MC  by  a', 
and  the  perpendicular  Mn 
be  drawn,  the  two  right 
angled  triangles  MnC  and 
MnT  will  irive 


fi  =  (x  ^  VnY  +  Mn. 
Subtracting  and  reducing,  we  have 


-  i^ 


4.rPrt. 


But  the  right   angled    triangle    MPw,    in    which   the   angle 
MPn  =   2  a',     gives 


hence 


Vn  =:  y  cos  2  a', 


<«  =   4^y  cos  2a^  =  4;?^  cos  2  a'. 


The  second  member  of  this  equation  is  constant,  and  will  there- 
fore be  the  same  for  any  position  of  the  point  M.  If  this  point  be 
placed  at  the  vertex  A,  the  half  diameter  will  be  CA  =:  a,  and 
the  half  tangent     AL  ==  h ;  hence 


192  INDETERMINATE    GEOMETRY. 

a^  —  i«  =  im  cos  2a' ; 
whence 

a'2  —  /2  ^  a2  -  5«. 
But,  Art.  (158),  we  have 

a'2  -  6'«  =  a«  -  J«, 
therefore,  we  must  have 

t  —  b'  or  n  =   26'; 

that  is,  if  a  tangent  he  drawn  at  any  point  of  the  hyperbola^  the  part 
intercepted  between  the  asymptotes  will  be  equal  to  the  conjugate  of 
the  diameter  passing  through  the  point  of  contact. 

Since  the  line  E'D'  is  equal  and  parallel  to  TS  =  QO,  it 
follows  that  the  figure  QOST  is  a  parallelogram,  and  that  the  ver- 
tices of  any  parallelogram  described  on  a  set  of  conjugate  diameters, 
will  lie  on  the  asymptotes. 


up     THE     POLAR     EQUATIONS  OF     THE     ELLIPSE     AND 
HYPERBOLA. 

166.     If  in  equation  (e).  Art.  (105),  we  substitute  the  values  of 
X  and  y  from  formulas  (2)  of  Art.  (69), 

a;  =  a'   -f  r  cos  v,  y  z=z  b'  ■\-  r  sin  v, 

we  shall  obtain  after  reduction 

(a«  sin*  V   +  Z»«  cos**  v)r^  +  2{a%'  smv  -{-  bH'  cos  v)r 

+  a%'^  +  6V2  -   a%^  =  0 (1), 

for  the  general  polar  equation  of  the  ellipse. 

By  changing  b^  into  —  i^,  this  will  become  the  general  polaf 
equation  of  the  hyperbola. 


k 


INDETERMINATE    GEOMKTRY.  I  Jo 

By  assigning  particular  values  to  a'  and  h',  in  the  alove  equa"- 
lion,  the  pole  may  be  placed  at  any  point  in  the  plane  of  the 
curve. 

First.     If  the  pole  is  on  the  curve,  we  must  have,  Art.  (109), 

a^'i  4-  5V2  —  a%^  =   0, 
and  the  equation  reduces  to 

[{a^sm^v+  b^co&^v)r  +   2{a^' smv  +  ^V  cos v) jr  —   0, 
which  may  be  satisfied  by  placing     r  ==   0,     or 
(a^sin^y   +  h^  cos^v)r  +   2(a'^^»' sinv  +   &V  cost;)   =   0 (2). 

The  pole  being  on  the  curve,  one  value  of  r  is  necessarily  equal 
to  0,  and  the  other  deduced  from  equation  (2),  will,  for  each  value 
of  V,  give  the  distance  from  the  pole  to  the  second  point,  in  which 
the  radius  vector  cuts  the  curve,  Art.  (VO). 

If  this  second  value  of  r  becomes  0,  the  radius  vector  will  he- 
come  tangent  to  the  curve,  and  equation  (2)  will  reduce  to 

a?b'  sin  v  -f  h^a'  cos  v  =   0, 

or 

sin  V         .  h^a' 

=  tang  V  =   —  , 

cos  V  a^b' 

as  it  should  be.  Art.  (128). 

Fur  the  hyperbola,  we  shall  have  the  same  discussion  and  re- 
sult, except  that  —  b^  takes  the  place  of  6*. 

Second.    If  the  pole  be  placed  at  the  centre,  we  have 

a'  =   0,  b'  =  0, 

which  reduces  equation  (1)  to  • 

(a^  sin«  V  -\-  b^  cos"  v)r^  —   a«6«   =   0 ; 

whence 

13 


104  INDETERMINATE    QEOMETKr. 


r  =   ±  J      .        "'^^ (..). 

*   a^  sin^  V  +  6*  cos*  v 

The  second  value  of  r  is  negative  for  all  values  of  v,  and  there- 
fore gives  no  point  of  the  curve,  Art.  (69). 

The  first  value  is  positive  for  all  values  of  v^  and  as  v  varies 
from  0  to  360°,  will  give  all  points  of  the  curve. 

^ ^  If    V  =   0,     sin  V  =  0,     cos  v  =  1, 

1^'       and  r  reduces  to 


A.  C      p  I"  B  r  =  a  =■  CB. 


If    V  =   90°,     sin  v  =   1,     cos  V  =  0,     and 

r  =  h  =CD. 

If  in  the  first  value  of  r,  (3),  we  put  for     sin*  v    its  value 
j<_  1  —  cos*  r,     divide  both  terms  of  the  fraction  under  the  radical 

viC^  iv\      sign  by  a*,  and  then  place 

a^    --    l^  » 
=  e*, 


e  representing  the  eccentricity  of  the  ellipse,  Art.  (118),  we  shall 
obtain 

h 
r  = . 


Vl   —  c*  cos*  V 
For  the  hyperbola,  equation  (3),  becomes 


=  ±\/ -^ ; 

^    a*  sin*  V  —   &*  cos*  v 


the  second  value  of  which  is  negative  for  all  values  of  v. 

The  first  value  is  positive  but  imaginary,  unless  the  denomina^ 
tor  is  negative  which  requires 

a*  sin*  V  —  t*  cos*  u  <  0,         or        tang  v  <  ±  -  . 

a 


INDETERMINATE    GEOMETRl. 


195 


If    v  =  0,     we  have, 
08  above 

r  =  a  =  CA. 

As  V  increases  from  0, 
:^    the  denominator  will  be 
negative  until 


a'  sin**  V  =  b^  cos'*  v,  tang  i»  =  _ , 

a 
when  the  value  of  r  will  be  infinite,  in  which  case     v  =  LCA, 

and  the  radius  vector  coincides  Avith  the  asymptote  CL,  Art.  (161). 

As  V  increases  beyond  this  value,  a^  sin^  v  becomes  greater  than 

b*  cos*  V,  and  r  will  be  imaginary  until 


tangv  =   —  _, 
a 


m  which  case     v  =  ACL"     and  the   radius    vector    coincides 
with  CL". 

When    V  =  180°,  we  have 

r  z=  a  =   CB, 

and  as  v  still  increases,  we  shall  continue  to  have  real  values  for  r 
until  it  coincides  with  CL'",  when  tang  v  again  becomes  equal  to 

- ,  and  from  this  point  the  values  of  r  will  be  imaginary  until  the 
a 

radius  vector  coincides  with  CL',  when  they  again  become  real  and 

continue  so  to     v  =  SQO°. 

The  first  value  of  r  thus  gives  all  the  points  in  both  branches 
of  the  hyperbola. 

By  a  process  similar  to  that  pursued  in  the  ellipse,  the  first  value 
of  r  may  be  reduced  to 


V  e«  cos«  V  —   1 


196  INDETERMINATE    GEOMETRY. 

in   wliich  e   represents   the    eccentricity  of  the   hyperbola,  Art. 

Third,  ii  the  pole  be  placed  at  the  right  hand  focus,  for  which 

((W.  Ill)  a'  =   Va^  ~   b^'  =  c,                  h'  =  0, 


equation  (1),  becomes 


\f 


(^ 


(a*  sin*  V  +  62  cos*  v)r^  +  26«c  cos  vr  —  h^  —  0.      "- 

If  for  sin*  v  we  put  its  value     1   —  cos*  v,     and  for     a*  —  6* 
its  value  c*,  this  equation  reduces  to 

-l?)  (a*  —  c*  cos*  vV*  4-  26*ccosvr  =  6*, 

from  which 


—    6*C  COS  V         ^     ^ 

'*  ~  a*  -  c*  cos*  v  ^  ^ 

/_ 

a* 

6*                      6'c*  cos*  V 
—  c*cos*  V        (a*  —  c*  cos*  vY 

or  reducing 

—  6*ccosi;  =fc  aJ* 

=h  a5*  —   6*c  cos  v 

^ : r~. = r  » 

a*  —  c*  cos*  v  (a  -h  c  cos  v)  (a  —  c  cos  v  ) 

whence,  the  two  values 


a  -f  c  cos  V  «  —  c  cos  V 

Since  for  the  ellipse 


c  =    Va*  —   6*  <  a         and         cos  v  <   1, 

the  second  value  of  r  is  always  negative  and  must  therefore  bo 
rejected. 

As  V  varies  from  0  to  360°,  the  first  value  of  r  will  bo  positive, 
and  give  all  points  of  the  ellipse. 

For  the  hyperbola,  expressions  (4)  and  (5)  become 

r   =  -^^ (6),  t   =  — ^^- (r). 

Q,  •\-  c  COS  V  a  —  c  cos  v 


INDETERMINATE    GEOMEl  IIY, 


197 


The  first  value  of  r  will  be  positive  whenever  the  denominator 
is  negative.     But  this  can  not  be  unless  cos  v  is  negative  and  nu- 
merically grcAter  than  _ .     Every  value  of  v,  beginning  with   0, 
c 

will   then  make  r  negative  until 

a                       a 
cos  V  =   —  -  =    —  — , 

when  the  radius  vector  will  be  parallel  to  the  asymptote  CL",  Art. 
(161)^  and  r  will  be  infinite.  As  v  now  increases,  cos  v  will  in- 
crease numerically  until  v  =  180°,  when  cos  v  =  —  1, 
and 


5* 


c  —    a 


which  is  positive,  and  gives  the  vertex  B.     As  v  increases  from 
this  point,  cos  v  will  di-        ;^  \ 

minish  numerically  and 
r  will  be  positive  until 
we  again  have 


cos  V  =z    —  _ , 
c 

when  r  =  CO ,  and 
the  radius  vector  becomes  parallel  to  the  asymptote  CL'".  All 
values  of  v  not  included  within  these  limits  will  make  the  first 
value  of  r  negative  and  give  no  points  of  the  curve.  Thus,  it  ap  - 
pears  that  this  value  of  r  gives  all  the  points  in  the  left  hand 
branch  of  the  hyperbola,  and  no  others. 

The  second  value  of  r  will  be  positive,  when  the  denominator  is 
positive.     Commencing  with     v  =  0    cos  r  =  1,     we  have 


r  = 


6« 


198  INDETERMINATE    GEOMETRY. 

which  is  negative.     As  v  increases,  cos  v  diminishes,  and  r  will  re* 
main  negative  until     a  =  c  cos  v,     when 


a 
cos  V  =   _  = 


Va3  +    h^ 


r  reduces  to  infinity,  and  the  radius  vector  takes  the  position  FR, 
parallel  to  the  asymptote  CL.     As  v  increases  from  this  point,  r 
will  be  positive  imtil  it  takes  the  position  FR'  parallel  to  CL'. 
When     v  =   90°,     cos  v  =  0     and 


=  ^  =  FM.        C^^  "0 


r 

a 


When     V  —  180°,     cos  v  =   -—   1,    and 

r   =z  =  FA. 

a   -\-  c 

The  second  value  of  r,  therefore,  gives  all  the  points  in  the  right 
hand  branch,  and  no  others. 

If  in  expressions  (4),  (6)  and  (7),  we  put  —  c  for  c,  the  pole,  in 
each  case,  will  be  placed  at  the  left  hand  focus. 

If  the  eccentricity  of  an  ellipse  be  denoted  by  e,  we  have,  Art. 
(118), 


e  z= 


»2     _     ^        _ 


a^  -  *a 


a  a"  a* 

from  which,  we  deduce 

c  =  ae,  h^  =  a%l    -  e^) (8). 

Substituting  these  values  in  expression  (4),  we  find 

a(l   -  e«) 
\  -\r  e  cos  V 


INDETERMINATE    GEOMETRY.  199 

which  expresses  the  value  of  r  in  terms  of  the  eccentricity  of  th« 
ellipse. 

For  the  hyperbola,  Art.  (119),  we  have 

c  ■=  ae,  -   68  =  a\\  —  e») (9). 

These  values  in  expressions  (6)  and  (V),  give 

^  ^     a(l   ~  e^)  ^  ^.  ^   _     «(!   -  e^)  ^ 

1  +  e  cos  i;  1  —  e  cos  v 

in  terms  of  the  eccentricity  of  the  hyperbola. 

From  equations  (8)  and  (9)  we  deduce  the  numerical  value 

a{\   -.«)  =  ?;  ^fW"'% 

hence,  the  numerator  of  each  of  the  above  values  of  r  is  equal  tc  i*^*^ 
one  half  the  parameter  of  the  curve,  Art.  (149) ;  as  is  also  the  case  "    | 
in  the  parabola.  Art.  (104).  ^^ 


^ 


DISCUSSION     OF     THE     GENERAL     EQUATION     OF     THE      .,  j 

SECOND     DEGREE.  CrtizL 

167.     Every  equation  of  the  second  degree  between  two  varia-    /x/^l.^ 
bles,  must  be  a  particular  case  of  the  most  general  form  " 

ay*  +  hxi/  +  cx^  -^  dij  +  ex  +  f  =  0 (1), 

which,  by  assigning  particular  values  to  the  constants  a,  b,  c,  &c., 
may  be  made  to  represent  every  line  of  the  second  order.  Art.  (33). 
Although  there  are  six  terms  in  the  above  equation,  and  ap- 
parently six  arbitrary  constants,  yet  it  must  be  observed  that  both 
members  of  the  equation  may  be  divided  by  the  coefficient  of  either 
term,  as  a,  thus  reducing  it  to  the  form 

y«  +  b'xT/  +  c'x^  +  d'y  +  e'a;  +  /  =  0, 


200 


INDETERMINATE    GEOMETRY. 


in  which  there  are  but  five  constants,  and  to  which  we  can  assign 
but  five  arbitrary  conditions. 

In  order  then  to  estimate  the  number  of  arbitrary  constants  in 
any  gener^il  equation,  or  equation  given  in  form  only,  we  divide 
l)y  the  coefficient  of  one  of  the  terms,  and  then  count  the 
number  of  different  coefficients  remaining.  This  will  indicate  the 
number  of  arbitrary  conditions  which  the  given  equation  may  be 
made  to  fulfil. 

In  commencing  the  discussion  of  equation  (l),  we  may  regard 
the  axes  of  co-ordinates, to  which  the  line  represented  by  it  is  re- 
ferred, as  at  right  angles  ;  for  if  they  were  oblique,  a  change  of 
reference  might  be  made  by  means  of  formulas  (5),  Art.  (67),  and 
a  new  equation  of  precisely  the  same  form  would  evidently  result. 


168.     By  solving  equation  (1),  of  the  preceding  article,   with 
reference  to  y,  we  obtain 


y  = 


hx  -]-  d       \ 


2a 


2a 


V(6«-4«c)a;3-f.  2{bd~-2ae)x+d^-4:af...{\\ 


from  which  we  may  readily  construct  the  line  by  points  as  in  Art. 
(22).  Each  value  of  x  which  will  make  the  quantity  under  the 
radical  sign  positive,  will  give  two  real  values  for  y,  and  coiTespond 
to  two  points  of  the  curve.  These  points  may  be  constructed  by 
laying  off  from  A  as  an  origin,  the  assumed  value  of  a:,  as  AP ;  at 
P  erect  the  perpendicular  PM,  on  which  lay  off 


PR  =   - 


hx  ■\-  d 
2a 


from  R  lay  off  RM'  in  the 
positive  direction  of  the  or- 
dinates,  and  RM  in  the  ne- 
gative, each  equal  to  the 
second  part  of  the  value  of 


INDETERMINATE    GEOMETRY.  201 

y ;  PM'  will  be  represented  by  the  first,  and  PM  by  the  second 

value  of  y,  and  M'  and  M  will  be  the  corresponding  points  of  the 

curve. 

Since  the  point  R  is  midway  between  the  two  points  M  and  M', 

it  follows  that  the  chord  MM'  is  bisected  at  R.     But  since  the 

points  R,  r,  (fee,  are  constructed  by  laying  off  the  different  values 

of  the  expression 

hx  -f  d 


it  follows  that  they  must  all  lie  on  the  right  line  whose  equation  is 

hx  -\-  d  hx  d 

y  —   —  ! —  ,  or         y  z=   -^  —  —  — . : 

2rt  2a         2a 

hence,  this  line  will  bisect  all  chords  drawn   parallel  to  the  axis  of 
Y  ;  it  is  therefore  a  diameter  of  the  curve,  Art.  (100),  and  may  at 

once  be  constructed  by  laying  off    AA'  =   —  — ,     and  through 

2a 

A',  drawing  the  line  A'R,  making  with  AX  an  angle  whose  tan- 
gent is     —    _ ,     Art.  (26). 
2a 

Hence,  if  an  equation  of  the  second  degree  he  solved  with  reference 
to  y,  the  first  member  placed  equal  to  that  part  of  the  second,  which 
does  not  contain  the  radical,  will  give  the  equation  of  a  diameter  bi- 
secting chords  parallel  to  the  axis  ofY. 

If  the  equation  be  solved  with  reference  to  x,  a  similar  dis- 
cussion will  show  that,  the  first  memher,  placed  equal  to  that  2^ctrt 
of  the  second  which  does  not  contain  the  radical,  will  give  the  equor 
Hon  of  a  diameter  bisecting  chords  parallel  to  the  axis  ofX. 


169.     If  in  equation  (1)  of  the  preceding  article,  we  place 
bd  —  2ae  =  m,  d*  —   ^af  =  n, 

\i  becomes 


202  INDETERMINATE    GEOMETRY. 


y  =  _  ^±±J:  rt  —V(b'  -  4ac)x'  +  2mx  +  n (1). 


Let  us  now  change  the  reference  of  the  points  of  the  curve  to 
a  new  set  of  axes,  of  which  the  diameter  A'X'  is  the  new  axis  of 
abscissas ;  the  new  axis  of  ordinates  being  parallel  to  the  prim- 
itive axis  of  Y.  In  the  formulas  (3),  Art.  (6*7),  we  must  then 
have 


d'  =.  90<=*,     cos  a'  =  0,     sin  a'  =  1,     tang  a  =  —  — , 


and  the  formulas  become 


a'  +  X'  cos  a,  y  =  &'  +  «'  sin  a  +  y'. 


Substituting  these  values  in  (1),  and  observing  that  since  the 
new  axis  of  X  is  a  diameter  bisecting  chords  parallel  to  the  new 
axis  of  Y,  each  value  of  x'  in  the  new  equation  must  give  two 
values  of  y  equal  with  contrary  signs,  and  therefore  in  the  re- 
sulting value  of  y',  the  part  independent  of  the  radical  must  dis- 
appeai",  we  have 


±   —  V  (6-  —  4ac)  {a'-^x'  cos  a)^4-  2m  (a'  -f  «'  cos  a)  •+- », 


INDETERMINATE    GEOMETRY.  203 

or  squaring,  clearing  of  deuonainators  and  developing, 


4aV"  =  (^'  —  ^ctc)  cosVa;'^  +  2  cos  a  [(6*  —  4ac)  a'  +  mjx'  + 
(b'  -  4ac)  a"  +  2wia'  +  w (2). 


Since  a',  in  this  equation,  is  arbitrary,  we  may  give  it  such  a 
value  as  to  make 


(h'-iac)a'  +  m  =  0,      or     a' =  __-^ . . . .  (3), 


in  which  case  equation  (2),  after  transposing  the  first  term  of  the 
second  member  to  the  first,  becomes 

iay  -  {b'  —  4ac)  cos^oa;'*  =  (6«  —  iac)  a''  +  2ma'  +n (4) 


If  i*  —  4ac  is  negative  ;  the  essential  sign  of  the  second 
term  of  the  first  member  will  be  positive.  If  the  second  member 
is  also  positive,  the  equation  ■s^'ill  be  of  the  same  form  as  equation 
(e'),  Art.  (143),  and  will  therefore  represent  an  ellipse  referred  to 
its  centre  and  conjugate  diameters. 

If  the  second  member  is  negative,  the  equation  will  indicate 
that  the  sum  of  two  positive  quantities  is  negative,  and  can  be 
satisfied  by  no  values  of  x'  and  y'.  The  line  represented  by  it  ia 
then  said  to  be  imaginary,  ard  an  imaginary  ellipse  is  a  particular 
case  of  the  ellipse. 

If  the  second  member  of  the  equation  is  0,  it  will  indicate  that 


204  INDETERMINATE    GEOMETRY. 

the  sum  of  two  positive  quantities  is  equal  to  0,  and  can  be  satis- 
fied by  no  values,  except 

/  =0,  x'  =  0, 

which,  Art.  (16),  are  the  equations  of  a  pointy  also  a  particulaf* 
case  of  the  ellipse. 

If  6  =  0     and     a  =  c,    we  have 

.  tang  a  =  0,  cos  a  =  1, 

and  equation  (4)  will  reduce  to  the  form 

which  is  the  equation  of  a  circle,  Art.  (35),  another  particular 
COM  of  the  ellipse. 

If  h^  —  4.ac  is  positive,  and  the  second  member  negative, 
equation  (4)  will  be  of  the  same  form  as  equation  (/i'),  Art.  (148). 
If  the  second  member  is  positive,  the  signs  of  all  the  terms  may 
be  changed  and  it  will  still  be  of  the  same  form,  x'  having  the 
place  of  y,  and  y'  the  place  of  a;,  Art.  (108).  In  either  case,  it 
will  therefore  represent  an  hyperbola  referred  to  its  centre  and 
conjugate  diameters. 

If  the  second  member  is  0,  the  equation  may  be  solved  with 
reference  to  y"^,  and  will  take  the  form 

y'2  -  rV^^  or  y'  =  ±  r'x\ 

representing  two  right  lines  which  intersect;  a  particular  case 
of  the  hyperbola. 

If  6  =  0     and     a  =  —  c,    the  equation  takes  the  form 

y"  -x''=  -  R^ 

the  equation  of  an  equilateral  hyperbola.  Art.  (112);  another 
particular  case  of  the  hyi^erhola. 

If  52  _  ^^g  _  Q^  ^jjQ  expression  (3)  will  be  infinite,  and  the 
value  of  a'  impossible;  but  under  this  supposition  equation  (2)  re 
duces  to 


INDETEKMINATE    GEOMETUV.  205 

4ay  =  ^m  cos  ckx'  +  2/?ia'  +  ^ (6), 

in  which  we  can  assiiju  to  a'  such  a  vahie  as  to  make 


o 


Ima'  +  n  =  0,  or  a'  =  -- — (6)  ; 


2m 


and  equation  (5)  reduces  to 


.  o  /o       «                                 /o       2m  cos  a 
4a''y  -^  =  2;?i  cos  ax'      or      v     =  --5 —  x' 


which  is  the  equation  of  a  parabola  referred  to  a  diameter  and 
tangent  at  its  vertex.     (Equa.  7,  Art.  99.) 

If  m  =  0,  expression  (6)  will  be  infinite,  and  the  vakie  of  a' 
impossible ;  but  in  this  case  equation  (5)  becomes 

4aV'.=  w,         or  y'  =   dt  1-^n, 

2a       ' 

x'  being  indeterminate^  and  will  represent  tivo  right  lines  parallel 
to  the  axis  of  X',  when  n  is  positive ;  one  right  line  which  coincides 
with  the  axis  of  X'  when  ti  =  0,  Art.  (21) ;  and  two  imagi- 
ry  right  lines  when  n  is  negative.     These  are  particular  cases  of 


•  ^  the  parabola 

^>/)r   no 


-^J^  170.  The  above  discussion  evidently  depends  upon  the  fact 
that  the  given  equation  contains  the  second  power  of  y,  or  that  a 
is  not  0. 

If  a  =  0  and  c  is  not,  the  equation  may  be  solved  with  re- 
ference to  ar,  and  the  same  results  deduced  in  precisely  the  same 
manner. 


200  INDETERMINATE    GEOMETRY. 

If    a  =  0     and    c  =  0,     and  b  is  not,  the  general  equa* 
tion  takes  the  form 

bx7j  +  dy  +  ex  -\-  f  =z   0 (1). 

Let  us  now,  by  the  aid  of  the  general  formulas,  Art.  (67), 
X  =  a'  -\-  x'j  2/  =  6'  -f  y', 

change  the  origin  of  co-ordinates,  without  changing  the  direction 
of  the  axes.     "VVe  thus  obtain 

bx'7/'  +  {a'b  +  dy/  +  {b'b  +  e)x'  +  a'b'b  +  I'd  -ha'e  +  f  =0....(2). 

In  this  equation  we  have  two  arbitrary  constants,  a'  and  6',  and 
may  therefore  assign  such  values  to  them  as  to  give 

a'b  +  d  =  0  b'b  +  e  =  0, 


,  d  jf  e 

a    _    ~  ^,  ""   ""  T* 

Substituting  these  values  in  equation  (2),  it  reduces  to 

b^,/  _  I  +  /  =  0,       or        .y  =  it^, 

which,  since  the  axes  of  co-ordinates  are  at  right  angles  to  each 
othei,  is  the  equation  of  an  equilateral  hyperbola  referred  to  its 
centre  and  asymptotes.  Art.  (161).  Equation  (1)  then  represents 
the  same  hyperbola,  referred  to  two  right  lines  parallel  to  its 
asymptotes. 

If  a  =  0,  6  =  0,  c  =  0,  the  equation  ceases  to  be  an 
equation  of  the  second  degree. 

From  the  previous  discussion,  we  conclude,  that  every  equation 
of  the  second  degree  between  two  variables  represents  one  of  the  conic 
sections,  that  is,  either  a  jyarabola,  an  ellipse  or  hyperbola,  or  one  of 
their  particular  cases. 


INDETERMINATE    GEOMETiiV.  207 

A  parabola  when  h^  —   4ac  =   0. 

An  ellipse  when  i*  —   4ac  <  0. 

An  hyperbola  when  b^  —  4ac  >  0. 

TJte  parabola  when  6*  —   4ow  =  0. 

IVI.  Under  this  supposition,  the  value  of  y,  equation  (1),  Art. 
(169),  reduces  to 

hx  ■{■  d    ^    1     , 

y  = ^^  =^  ^^2,n^  +  « -O)- 

Every  value  of  a?,  wliich  will  make  the  quantity  under  the  radi- 
cal sign  positive,  will  give  two  real  values  of  y  and  two  correspond- 
ing points  of  the  curve. 

The  value  of  a?,  which  makes  this  quantity  0,  will  give  two  equal 
values  of  y,  the  two  corresponding  points  unite,  and  the  ordinate 
produced  is  tangent  to  the  curve,  Art.  (35). 

Every  value  of  rr,  which  makes  the  quantity  under  the  radical 
sign  negative,  gives  imaginary  values  for  y  and  no  points  of  the 
curve. 

If  we  place 

2mx  4-  n  =  0,  we  have  x  —   —  — , 

which  is  the  only  \  iilue  of  x  that  will  reduce  the  quantity  under 
the  radical  sign  to  0.  Denoting  this  value  by  a;',  the  value  of  y 
may  be  written 

bx  +  d    ,     1     , 

y  =  — Trjr  "^  5^V2m{.^  -  X') (2), 

•inoe 

2mx  4-  w  =  277i  (.r   +  ^). 
^  2m! 


r  — 


208 


INDETERMINATE    GEOMETRY. 


Now,  if  m  is  positive  ;  whether  x'  be  positive  or  negative,  every 
vahie  of  x  y-  x'  will  give  two  real  and  unequal  values  for  y  ; 
X  ^=  x'  will  give  two  equal  values  ;  and  every  value  of  a;  <  a;' 
will  give  imaginary  values.  Hence,  the  curve  extends  indefinitely 
in  the  direction  of  a;  positive,  is  tangent  to  the  ordinate  PV,  which 
corresponds  to  the  abscissa  x'^  and  has  no  points  on  the  left  of  this 
ordinate,  as  indicated  by  the  full  line  in  the  figure. 

If  m  is  negative  J  every  value  of  x  ^  x'  will  give  imaginary 
values  for  y;     x  =  x'     will  give  equal  values,  and      x  <  x' 

will  give  two  real  and  un- 
equal values.  Hence,  the 
curve  is  limited  in  the  di- 
rection of  X  positive  by  the 
produced  ordinate  PV,  and 
extends  indefinitely  in  the 
direction  of  x  negative,  as  in- 
dicated by  the  dotted  line  in 
the  figure.  Hence,  in  order 
to  obtain  the  limit  of  the 
curve  in  the  direction  of  the  axis  of  abscissas,  we  solve  its  equation 
with  reference  to  y,  place  the  quantity  under  the  radical  sign  equal 
to  0,  and  deduce  the  value  of  .r,  this  value  will  he  the  abscissa  of  the 
limit,  lay  it  off  and  through  its  extremity  draw  a  line  parallel  to  the 
axis  of  Y,  it  will  be  the  limit ;  and  this  limit  will  be  tangent  to  the 
curve  at  the  vertex  of  that  diameter  which  bisects  the  chords  par- 
allel to  the  axis  of  Y. 

K  the  coefficient  of  x  under  the  radical  sign  is  positive,  the  curve 
will  lie  entirely  on  the  right  of  this  limit ;  if  negative,  on  the  left. 
By  solving  the  equation  with  reference  to  a*,  we  may,  in  a  similar 
way,  construct  the  limit  in  the  direction  of  the  axis  of  Y. 
If    m  =  0,     the  value  of  y,  equation  (1),  becomes 


hx  ■\-  d 
2a 


2a 


Vuy 


or 


INDETERMINATE    GEOMETRY.  209 

hx  d  yn  hx  d  yn 

tf  z=  —  —  —  —  +  — ,         y  =  —  —  —  —  —  —  1 

2a         2a  2a  2a         2a  2a 

the  equations  of  two  parallel  straight  lines,  Art.  (28),  when  n  is 
positive  ;  which  reduce  to  (me  straight  line,  when  9i  =  0  ;  and 
to  two  imagina.ry  parallels,  when  n  is  negative,  as  seen  in  Art. 
(169).  Hence,  an  equation  of  a  parabola  being  solved  with  re- 
ference to  either  variable,  if  the  quantity  under  the  radical  sign 
is  a  positive  constant,  the  equation  will  represent  two  parallel 
straight  lines. 

If  this  quantity  is  0,  or  the  radical  disappears,  the  equation  will, 
represent  one  straight  line.  If  this  quantity  is  a  negative  constant, 
the  equation  will  represent  two  imaginary  parallels. 

It  may  be  remarked,  that  in  the  first  case,  the  line  whose  equa- 
tion is 

hx    ■\'    d 

bisects  all  chords,  terminated  in  the  two  Hnes  and  parallel  to  the 
axis  of  Y,  and  therefore  strictly  fulfils  the  condition  of  a  diameter, 
Art.  (100). 

In  the  second  case,  the  line  represented  by  the  equation  is  the 
diameter  itself. 

In  the  third  case,  the  diameter  may  be  constructed  while  the 
lines  do  not  exist. 


172.  By  solving  the  equation  with  reference  to  ar,  we  find  for 
the  equation  of  the  diameter  which  bisects  all  chords  parallel  to 
th<jaxisofX,  Art.  (168), 

x=    ^^JLAl;  whence  y  =   «   ?ff  _  1  ; 

2c      '  ^  6  6' 

but  since     6'  —  4ac  =  0,     we  have 
14 


210  INDETERMINATE    GEOMETRY. 

^   _    2c 
2a  ~  y 


Jience  the  coefficient  of  x  in  the  above  equation  is  equal  to  the  co- 
efficient of  X  in  the  equation 


hx  d 

2a         2a' 


and  the  two  diameters  represented  by  these  equations  are  paral- 
lel, Art.  (28). 


173.  By  an  application  of  the  foregoing  principles  we  are  ena 
bled  to  represent  on  paper,  a  parabola  whose  equation  is  given, 
without  taking  the  trouble  to  determine  many  of  its  points. 

First,  find  the  points  in  which  the  curve  cuts  the  axes  of  co- 
ordinates, Art.  (2j^) ;  then  solve  the  equation  with  reference  to 
each  variable  in  succession,  and  construct  the  diameters  which  bi- 
sect the  chords  parallel  to  the  axes,  Arts.  (168),  (26) ;  then  con- 
struct the  limits  of  the  curve  in  the  direction  of  both  axes,  Art, 
(lYl) ;  and  draw  a  curve  tangent  to  these  hmits  at  the  points  at 
which  they  intersect  the  diameters  and  through  the  points  first  de- 
termined, taking  care  to  make  it  symmetrical  with  respect  to  both 
of  the  diameters. 

Examples. 

First,  when  m  is  not  0. 

1.      y2   -    2x?/   +   .f2   —   y    -}-    2^   -    1    =    0 (1). 

By  comparing  this  with  the  general  equation.  Art.  (167),  w^ 
Bee  that 

o  =  1,     d  =  ..-  2,     f  -  1,  6«  -  4ac  =  4  -  4  =  0  ; 


INDETERMINATE    GEOMETRY. 


211 


henco,  the  curve  is  a  parabola. 
Making     y  =   0,     we  obtain 

a;*  +  2:r—    1    =   0;  x  =    —   1   ±:   V^. 

Assuming  any  line  as  a  unit  of  measure  and  laying  off 


AB  =   -   1   +   V  2, 

K 

V^ 

pC 

AB'  =   -  1   -    V^ 

// 

Vl 

^? 

we  have   the   points   in 
which    the    curve    cuts 

,/^ 

/ 

/ 

/     / 

the  axis  of  X.    Making 

// 

X  =  0,     we  find 

/ 

'  =  \  *v/f 


and  may  thus  determine  the  points  C  and  C  in  which  the  curve 
cuts  the  axis  of  Y. 

Solving  the  given  equation,  first  with  reference  to  y,  and  then 
with  reference  to  ar,  we  have 


y 


2a;  +   1  1 


2  2 

a;  =   y—    Id:-/   —   y4-2 


V  -   4a;  +  6 (2), 

(3). 


The  equations  of  the  diameters  are 
2a;  +   1 


«  =  y  -  1, 


Nvhich  represent  the  lines  DV  and  D'V. 

Placing  the  quantities  under  the  radical  signs  (2)  and  (3),  equal 
to  0,  we  deduce 


for  the  first, 


..5. 


212 


INDETERMINATE    GEOMETRY, 

for  the  second,  y  =   2. 


Laying   off     AP  =  _     and  drawing  the  line  PV,  it  must  be 
4 

tangent  to  the  curve  at  V,  and  since  the  coefficient  of  x  under  the 

radical  sign  is   —  4,  the  curve  will  lie  on  the  left  of  this  tangent. 

Laying  off    AR  =   2,     and  drawing  the  line  RV,  it  will  be 

the  limit  in  the  direction  of  the  axis  of  Y,  and  the  curve  will  be 

represented  as  in  the  figure. 

2.     y^  —  Ixy  •\'  x^  ■\-  y  —  2x  =  0. 


3.  y2  +  2a!y  +  a:2  —  2y  —  1  =  0. 

4.  y«  —  Ixy  +  a;2  —  22/  —  1x  =  0. 

5.  y^  4-  2a:y  -^  x^  +  2y  =  0. 


y' 


^xy  ■\'  x^  ■\-  X 


Second^  when    m  =  0     and  n  positive. 


1.     y«  —  "Ixy  -f  .i;«  —  2y  +  2a:  —  1  =  0, 


2.  2/3  _  2ary  +  a;*  —  1  =  0. 

3.  y8  +  4.ry  +  4ar'»  +  4  =  0. 


/ 

Thii'd^  when  m  =   0,     ri  =  0. 

1.  y^  —  Ixy  +  a;**  +  2y  —   2a;  4-   1    =  0. 

2.  y«  —  Axy  -f  4.r2  =   0. 

Fourth,  when  m  ■=  0,     am?  w  negative. 

1.  y'*  +  2ary  +  .'<;«  +   1=0. 

2.  2/«  +  y  +   1    =   0. 


INDETERMINATE    GEOMETRY. 


213 


]  14.  If  it  is  required  to  construct  the  curve  with  accuracy ;  we 
may  first  solve  its  equation  with  reference  to  ?/,  construct  the 
diameter  and  determine  the  hmit  as  in  Art.  (1*71).  This  limit  is 
tangent  to  the  curve  at  the  point  in  which  it  intersects  the  diame- 
ter. Solve  the  equation  with  reference  to  x,  construct  the  diame- 
ter and  determine  the  limit  in  the  direction  of  the  axis  of  Y.  This 
is  also  tangent  to  the  curve  at  the  point  in  which  it  intersects  the 
diameter.  Since  these  tangents  are  parallel  to  the  co-ordinate 
axes  respectively,  they  are  perpendicular  to  each  other  and  inter- 
sect on  the  directrix,  Art.  (9*7).  Through  their  point  of  intersec- 
tion draw  a  line  perpendicular  to  either  diameter,  it  will  be  the 
directrix,  Art.  (100).  Join  the  two  points  of  tangency  by  a  chord, 
this  will  pass  through  the  focus.  Art.  (97).  With  either  point  of 
tangency  as  a  centre,  and  the  distance  to  the  directrix  as  a  radius, 
describe  an  arc,  it  will  cut  the  chord  in  the  focus,  Art.  (88). 
Through  the  focus  draw  a  perpendicular  to  the  directrix,  it  will  be 
the  axis,  and  the  curve  may  then  be  constructed  as  in  Art.  (88). 

To  illustrate,  let  us  recur  to  example  (1)  in  case  first,  of  the  pre- 
cedinor  article.  Ha\dn2: 
determined  the  limits  PV 
and  RV,  through  their 
point  of  intersection  S, 
draw  SO  perpendicular 
to  DV,  it  is  the  direc- 
trix ;  join  the  points  V 
and  V ;  with  V'E  de- 
scribe the  arc  EF  cuttinor 
W  in  F,  F  is  the  focus  through  which  the  axis  may  be  drawn 
parallel  to  DV. 

2%e  ellipse  when     b^  —  iac     is  negative. 


175.     The  value  of  y,  equation  (1),  Art.  (168),  may  be  put  un- 
der the  form 


214  INDETERMINATE    GEOMETRY. 


^  2a       "^  2a  V  ^  ^\^      ^  h'i  ~  Aac  ^  b^^4acj 

Those  values  of  x^  which  will  reduce  the  radical  to  0,  and  give 
equal  values  of  y,  will  evidently  be  obtained,  by  placing 

^,   +        2"^        +  __» =  0. 

b^  —   4ac         6*  —  4a€ 

Solving  this  equation,  and  denoting  the  least  value  of  x  by  x' 
and  the  other  by  x",  the  value  of  y,  may  be  put  under  the  form 

y  =  —  ^^_JJlA  -i-  JL  V(^,2  _  4ac)(x  -  x'){x  -  x") (1). 

These  roots  x'  and  ir"  may  be  real  and  unequal,  real  and  equal, 
or  imaginary. 

When  real  and  unequal.  For  every  value  of  a;  >  x"  the 
factors  x  —  x"  and  rK  —  a;'  will  both  be  positive,  their  pro- 
duct also  positive,  and  the  quantity  under  the  radical  sign  nega- 
tive. The  corresponding  values  of  y  will  therefore  be  imaginary, 
and  there  will  be  no  corresponding  points  of  the  curve. 

For  X  =  rr",  the  quantity  under  the  radical  sign  is  0,  the 
two  vMues  of  y  equal,  and  the  ordinate  produced  is  tangent  to  the 
curve  fvt  tlie  vertex  of  the  diameter  whose  equation  is,  Art.  (168), 

hx  ■\'  d 

^  2a 

For  every  value  of  a;  <  x"  and  >  ar',  the  two  factors  x  —  x*^ 
and  x—  x'  will  have  contrary  signs,  their  product  will  be  negative, 
and  tlio  quantity  under  the  radical  sign  positive,  and  there  will  be 
two  corresponding  real  values  of  y  and  two  points  of  the  curve. 

For  X  =  a;',  the  quantity  under  the  radical  sign  again  bo- 
comLS  0,  and  the  ordinate  will  be  tangent  to  the  curve  at  the  other 
vertex  of  the  diameter. 


INDETERMINATE    GEOMETRY. 


213 


For  every  value  of  a;  <  x'^  the  factors  x  —  x"  and  x  —  x' 
m\\  be  negative,  their  product  positive,  and  the  values  of  y  imagi- 
nary. 

Therefore,  if  two  distances  AP  and  AP',  represented  by  x'  and 
and  x"^  be  laid  off  on  the  axis  of  X,  and  through  their  extremities 
two  lines  be  drawn  parallel  to  the  axis  of  Y,  these  lines  will  be 
tangent  to  the  curve,  and  no  point 
of  the  curve  can  lie  without  them. 
Hence,  to  obtain  the  limits  of  the 
curve  in  the  direction  of  the  axis  of 
abscissas ;  we  solve  the  equation 
with  reference  to  y,  place  the  quan- 
tity under  the  radical  sign  equal  to 
0,  and  deduce  the  roots  of  the  equation,  these  ivill  he  the  abscissas  of 
the  limits ;  lay  off  these  abscissas,  and  through  their  extremities 
draw  lines  parallel  to  the  axis  of  ordinates,  they  will  be  the  required 
limits.  These  limits  will  be  tangent  to  the  ellipse  at  the  vertices 
of  the  diameter  which  bisects  all  chords  parallel  to  the  axis  of  Y. 

By  solving  the  equation  with  respect  to  x,  we  may  obtain,  in  a 
similar  way,  the  limits  in  the  direction  of  the  axis  of  Y. 

If  the  roots  x'  and  x"  are  equal,  we  have 


{x  -  x'){x  -  x")  =   (a:  -  x'Y, 
and  the  value  of  y  reduces  to 


hx  ■\-  d    ,    X  —  x'     , 

y  = r ±  — ^/h^  -   4ac, 


2a 


2rt 


which  will  evidently  be  imaginary  for  every  value  of  x  except 
X  =  x\    and  this  gives  for  the  corresponding  value  of  y,  denoted 

byy', 


/=  - 


Ix'  4-  d 
2a 


9.16 


INDETEVMINATE    GEOMETRY. 


K 


y'  and  x'  are  then  the  co-ordinates  of  a  single  point,  to  "whicli  the 
ellipse  in  this  case  reduces,  Art.  (168). 

If  the  roots  x'  and  x"  are  imaginary,  the  product     {x  —  x') 
(x  —  x")     will  be  positive  for  all  values  of  a;  *  ;  hence,  every 
value  of  a:,  in  equation  (l),  will  give  imaginaiy  values  for  y,   and 
there  can  be  no  points  of  the  curve,  which  is  said  in  this  case  to  he 
\  a^jMmaginary,  Art.  (1G8). 


■5^ 


1*76.     An  equation  of  an  ellipse  being  given,  the  curve  may  be 
well  represented  by  following  the  rule  laid  down  in  Art.  (1'73). 


Exam,ples. 
First,  when    x'  and  x"    are  real  and  unequal. 


y«  _   2xy   +   2a:«  -h   2y  —  ar  —    0, 

in  which   l/^  —  4ac  =  4  —  8 
and 


y  —  X  —  I  ±:  V—  x'^  —  X  -\-  \, 

x'  =  K?"  =  -  1  _  \/5, 
2         >'  4 


*  Note. — To  prove  this,  we  have  only  to  recollect  that  imaginary  roots 
always  enter  an  equation  in  pairs,  and  must  be  particular  cases  of  the 
general  form 

X  =i  ajr  h^ —  1, 

(he  factors  corresponding  to  which  are 

.r  —  (a  -f  h^  —  1)  and 

their  product  being 

z2  —  2ax  +  a'2  +  A2  _    X  —  of  Ac-  V^ , 


_(a-i/HT). 


which  is  evidently  positive  for  all  values  of  .r,  since  it  is  th-?  sum  of  two 
perfect  squares. 


INDETERMINATE    GEOMETRY. 

X"  =  AF  =  -  i  +  V-. 

2.  y«  —   2x7j  +  2x^  —   2y  —   2x  =   0. 

3.  y^  +   2xy  +   2x'^  —   2:c  =   0. 

4.  2y«  —   22:y  +   3^2  ^   2y  +  re  —   1   = 
Second^  when     x'  and  x"     are  real  and  equal. 

1.  2/8  —   2xy  +   2^2  —   4;r   +   4   =   0. 

2.  ya   +   a;2   _    2^   +    1    =   0. 
Third,  when     x'  and  x"     are  imaginary. 

1.  2/2    ^    a;y    ^    ^2    _}.    a;    ^    y    _j.    1    =    0. 

2.  y*  +   a;*  +  2a;  +   2   =   0. 


217 


17Y.     In  order  to  construct  the   curve  accurately ;  we  solve  the 
equation  with  reference  to  y,  con-  r 

struct  the  diameter  and  determine 
the  abscissas  of  the  limits  as  in 
Art.  (l7o).  Substituting  these  in 
either  the  equation  of  the  curve  or 
diameter,  we  find  for  the  ordinates 
of  the  vertices  V  and  Q, 


,  hx'  ■\'  d 

y'  =  - 


y"  = 


hx"   +  d 
2a        '  '  2a. 

Substituting  these  in  expression  (2),  Art.  (17),  we  have 


^  4a2  ■  ^  2a 

Since  this  diameter  bisects  chords  parallel  to  the  axis  of  Y,  ita 


INDETERMINATE    GEOMETRY. 


conjuQ:ate  will  be  V'Q',  passing  through  the  centre  C  and  parallel 
to  AY,  Art.  (143).  If  we  denote  the  abscissa  of  the  point  C  bj 
z,  and  substitute  it  in  equation  (1),  Art.  (iVo),  we  have  for  the 
corresponding  values  of  y,  P"V'  and  P"Q', 


bz  +  d  ^    1     , 

y  = ^^  ±  -V(6^  -  4.ac){z  -  x'){z  - 


The  difference  of  these  two  values  is  the  length  of  V'Q' ;  hence, 
^'Q'  =7  -Vlh^  -  4.ac){z  -  x'){z  -  x'% 

or  substituting  for  z  its  value,  which  is  evidently 

x'  +  cc" 

we  have 

V'Q'  =  '^-^^iac  -  lA 
2a 

The  length  and  position  of  these  two  conjugate  diameters  being 
now  known,  the  curve  may  be  constructed  as  in  Art.  (150). 

The  angle  V'CQ,  made  by  the  conjugate  diameters,  may  be 
readily  measured,  since  the  tangent  of  the  angle  CDP",  in  any 
position  of  the  diameter,  will  have  the  same  numerical  value  as 

tang  a,  and  therefore  be  equal  to    —  —     taken  with  a   positive 

2a 

gign  ;  whence,  by  a  reference  to  a  table  of  natural  sines,  &c^  CDP'' 

becomes  known,  and  since 

V'CV  =  90°  -  CDP", 

we  have 

V'CQ  =   180°  -  V'CV  =  90°  +  CDP". 

The  two  conjugate  diameters  and  the  angle  made  by  tliem 


INDETERMINATE    GEOMETRY.  219 

being  thus  known,  the  curve  may  be  constructed  as  in  Art.  (150), 
or  the  axes  as  well  as  the  angles  a  and  a'  may  be  determined 
from  equations  (1),  (2),  and  (3),  Art.  (157). 

The  Hyperbola  when     h^  —  4ac     is  positive. 
178.     Let  us  resume  the  value  of  y,  equation  (1),  Art.  (175), 

hx  +  d    ,     1     . L 

y  = ^^  ^  ^V(6«  -  4ac)(x  -  x'){x  -  X") (1), 

in  which,  we  must  remember  that  x'  and  x"  are  the  values  of  x 
obtained  by  placing  the  quantity  under  the  radical  sign,  in  the 
general  value  of  y,  equal  to  zero,  and  that  they  will  be  real  and 
unequal^  real  and  equal,  or  imaginary. 

When  real  and  uneqiial.  For  every  value  of  ar  >  x",  the 
factors  X  —  x"  and  x  —  x'  will  both  be  positive,  and  the 
quantity  under  the  radical  sign  positive.  The  corresponding 
values  of  y  will  therefore  be  real  and  unequal,  and  there  will  be 
two  corresponding  points  of  the  curve. 

For  x  =  x"  the  quantity  under  the  radical  sign  is  zero,  and 
the  corresponding  ordinate  produced  will  be  tangent  to  the  curve 
at  the  vertex  of  that  diameter  which  bisects  chords  parallel  to  the 
axis  of  Y,  Art.  (168). 

For  every  value  oi  x  <^  x"  and  >  x',  the  two  factors 
will  have  contrary  signs,  their  product  will  be  negative,  and  the 
corresponding  values  of  y  imaginary,  and  there  will  be  no  corres- 
ponding points  of  the  curve. 

For  X  =  x\  the  corresponding  ordinate  produced,  again  be- 
comes tangent  to  the  curve  at  the  other  vertex  of  the  above  di- 
ameter. 

For  every  value  of  a;  <  x\  the  factors  will  both  be  negative, 
their  product  positive,  and  the  corresponding  values  of  y  real. 

Therefore,  if  two  distances  AF  and  AP',  represented  by  x'  and 
x'\  be  laid  off  on  the  a\is  of  X,  and  through  their  extremities  two 


220 


INDETERMINATE    GEOMETRY. 


lines  be  drawn  parallel  to 
the  axis  of  Y,  these  lines 
"vvill  be  tangent  to  the  curve, 
no  point  of  the  curve  wil) 
lie  between  them,  and  the 
curve  will  extend  to  infinity 
in  both  directions  without 
them.  Hence,  we  obtain 
the  limits  of  the  hyperbola  in  the  direction  of  either  axis  of  co-ordi- 
nates in  the  same  way  as  described  in  Art.  (Ho). 

If  the  roots    x'  and  x"     are  equals  the  value  of  y,  equation  (1), 
as  in  the  corresponding  case  in  the  ellipse,  Art.  (1*75),  reduces  to 


hx  +  d 
2a 


X    —    X' 

2a 


^/h^   _   4ac, 


■which  will  evidently  be  real  for  every  value  of  x.     This  equation 

then  represents  two  right  lines  lohich  intersect,  and  to  which  the 

hyperbola  in  this  case  reduces. 

If  the  roots     x'  and  x"     are  imaginary,  the  product     [x  —  x') 

{x  —  x")     will   be  positive  for  all  values  of  x\    [see  note,  Art. 

(175)],  hence  every  value  in  equation  (1),  will  give  real  values 

for  ?/,  and  two  corresponding  points 
of  the  curve,  and  there  will  be  no 
limits  in  the  direction  of  .the  axis  of 
X,  as  was  to  be  expected,  since  the 
abscissas  of  these  limits  x'  and  x" 
are  imaginary.  It  also  follows,  that 
the  diameter  which  bisects  chords 
parallel  to  the  axis  of  Y,  has  no 

vertices,  or  does  not  intersect  the  curve,  which  must  be  as  repro- 

senied  in  the  figure. 


179.     An  equation  of  an  hyperbola  being  given,  the  curve  may 
be  well  represented  by  following  the  rule  laid  down  in  Art.  (1 73). 


INDETERMINATE    GEOMETRY, 


221 


Examples. 

First,  when    x'  and  x"     are  real  and  uneqv.al. 
I.     2/2  —  2xy  —  a;8  4-  2  =  0. 

iu  which 

J3_4ac  =  4  —  4xlX  — 1 

=  4+4  =  8, 

ftud 


y  =z  X  dt.   V2a;2  —   2. 
2.     ?/«  _  ar*  +  2a;  —  2?/  +  1  =  0.  / 

3.  ?/«  -\-  xij  —  2x^   +  ic  =    0. 

4.  if  —  2xy  —  a;2   —   2?/  +   2a;  +  3    =   0. 

Second,  when  x'  and  x"     are  real  and  equal. 

1  2/'  —   2a;8  +   2y  +   1   =   0. 

2.  2/8  —  a;«  =   0. 

3.  y'^  -\-  xy  —   2x^  +  Zx  —   1   =   0. 
Third,  when  x'  and  x"     are  imaginary. 

1.  y^  —   2xy  ^  x^  —   2  =  0. 

2.  3/2  +   2a-?/  —  a;«  +   2a;  4-  2y  —  1   =  0. 

3.  2/2  _  2a;y  —  a;'  —   2a;  —   2   =  0. 


180.  llie  curve  may  also  be  constructed  accurately,  by  first 
determining  the  length  and  position  of  two  conjugate  diameters, 
precisely  as  in  Art.  (1*77).     The  expressions  for  these  diameters 


--^  INDETERMINATE    GEOMETRY. 

will  be  the  same  as  those  determined  for  the  ellipse.  For  ihe. 
distance  cut  off  by  the  curve  on  the  one  which  bisects  chords 
parallel  to  the  axis  of  Y,  we  have 

/p/    2;"      


2a 


and  on  its  conjugate 


2a 


•v/4ac  —   h"^, 


the  first  of  which  will  be  real,  and  the  second  imaginary,  when  x' 
and  x"  are  real,  and  the  reverse  when  x'  and  x"  are  imaginary. 

In  this  case,  the  angle  VCD  [see  figures  in  Art.  (IV8)],  inclu- 
ded between  the  two  conjugate  diameters,  is  always  equal  to 
90°  —   CD  A.     But  we  know  that  tang  CDA  is  numerically  equal 

to     tan  or  a  =  — We  therefore  have 

^  2a 

tang  VCD  =  cot  a, 

from  which  the  angle  may  at  once  he  found,  and  then  the  curve 
be  constructed  as  in  Art.  (150),  or  the  axes,  together  with  a  and 
a',  may  be  found  from  equations  (1),  (2)  and  (3)  of  Art.  (158). 


OF    CENTRES    AND    DIAMETERS. 

•  181.     The  centre  of  a  curve  is  a  point,   through  which,  if  any 

straight  line  he  drawn,  terminating  in  the  curve,  it  will  he  bisected 
at  this  point. 

It  follows  from  this  definition,  that  for  each  point,  as  M,  of  a 
curve  which  has  a  centre,  there  will  be  another  corresponding 
point,  as  M',  on  the  oi)posite  side  of  the  centre  and  at  the  same 
distance  from  it.  If  therefore  the  origin  of  co-ordinates  be  placed 
at  the  centre,  the  co-ordinates  of  these  two  points  will  be  equal 


INDETERMINATE    GEOMETRY. 


with  contrary  signs  ;  that  is,  if  the 
co-ordinates  of  one  point  are  +  «' 
and  +  y'l  those  of  the  other  will  he 
—  x'  and  —  y',  and  the  equation 
of  the  curve  must  be  satisfied  by 
the  substitution  of  each  of  these 
sots  of  co-ordinates.  But,  this  can 
not  be    the   case,  unless    all  the 

terms  of  the  equation  containing  the  variables  are  of  an  even  de- 
gree ;  for  if  some  are  of  an  odd  degree,  the  signs  of  these  terms 
will  be  different  when  —  x'  and  —  y'  are  substituted,  from  what 
they  are  when  -f  x'  and  -f  y'  are  substituted,  while  those  of  an 
even  degree  will  remain  the  same.  It  is  evident  then,  that  the 
equation  can  not  be  satisfied,  in  both  cases. 

In  order  then  to  ascertain  whether  a  given  curve  has  a  centre, 
we  fii*st  examine  its  equation  and  see  if  all  its  terms  are  of  an  even 
degree  with  respect  to  the  variables.  If  they  are,  the  origin  of 
co-ordinates  is  a  centre.  If  they  are  not,  we  substitute  for  the  va- 
riables their  values  taken  from  the  formulas  (2),  Art.  (67),  and  see 
if  such  values  can  be  assigned  to  the  arbitrary  constants  a'  and  h' 
as  will  cause  all  the  terms  of  an  odd  degree  to  disappear.  If  so, 
the  curve  will  have  a  centre  at  the  new  origin,  and  the  values  of 
a'  and  b'  will  be  its  co-ordinates  when  referred  to  the  primitive 
system.  If  no  real  and  finite  values  can  be  thus  assigned,  the 
curve  will  have  no  centre. 


182.    To  apply  the  above  principles  to  lines  of  the  second  or- 
der, we  resume  the  general  equation 

ay^  -I-  hxy  ■\-  cx"^  -^  dy  -^  ex  ■\-  f  =  0, 

and  substitute  for  a;  and  y  their  values  taken  from  tho  formulas 
of  Art  (67). 


T  =  a'   -f  x\ 


y  -  h'   -\-  y'. 


224  INDETERMINATE    GEOMETRY. 

WB  tlius  obtain,  after  reducing,  and  denoting  the  sum  of  all  the 
terms  independent  of  tlie  variables  hjf. 

ay'^  +  bx'y'  +  cx'^  +  ('2ah'+  ba'+d)i/'-\-{2ca'-}-bb'+e)x'-\-f'=  0. 
Tlfe  terms  of  this  equation  will  all  be  of  an  even  degree,  if 
2ab'  +  ba'  -\-  d  =  0,  2ca'   +  bb'  +  e  =  0, 

which  give  for  a'  and  Z>',  the  values 

JK  ,         2ae  —  bd  ,,         2cd  —  be 

ft'-*    —    4ac  b^  —  4ac 

These  will  be  real  and  finite  when  b^  —  4:ac  is  not  zero,  from 
which  we  conclude  that  there  is  always  a  single  centre  for  each 
dlipse  and  hyperbola. 

When  &*  —  4ac  =  0,  and  the  numerators  are  not  zero, 
the  above  values  reduce  to  infinity  ;  from  which  we  conclude  that, 
in  general,  the  centre  of  the  parabola  is  at  an  infinite  distance,  or 
that  the  parabola  has  no  centre. 

If  6*  —  4ac  =  0  and  2ae  —  bd  =  0^  we  must  also 
have 


^ 


2cd  —   be  =  0, 


2ae 
for  by  substituting  in  this  the  value  of  c?  =  — -  ,    taken  from  the 

b 

last  expression,  it  becomes 

4ace         ,           »                           (iac  —  b^)e         ^ 
be  z=  Oj  or  i^ L-  =  0  ; 


hence,  in  this  case  the  two  values  of  a'  and  5'  both  become  -  ,    or 

0 

indeterminate  ;  from  which  we  conclude  that  there  is  an  infinite 
number  of  centres,  which  was  plainly  to  be  anticipated,  as  in  this 
case  the  parabola  reduces  to  two  parallel  right  hues,  Art.  (171), 


INDETERMINATE    GEOMETRY.  225 

and  any  point  of  the  diameter  midway  between  them  will  fulfil  the 
condition  of  a  centre. 


183.  A  diameter  of  a  curve  is  any  straight  line  which  bisects  a 
system  of  parallel  chords  drawn  in  the  curve,  Art.  (100).  * 

In  lines  of  the  second  order,  if  the  axis  of  X  be  a  diameter  and 
the  axis  of  Y  be  placed  parallel  to  the  chords  which  this  diameter 
bisects,  it  is  evident  that  the  equation  of  the  curve,  when  referred 
to  these  axes,  must  be  of  such  a  form  as  to  give  for  each  single 
value  of  Xy  two  values  of  y,  equal 
with  contrary  signs.  Thus  if  AX 
be  a  diameter,  taken  as  the  axis  of 
X,  and  x\Y  be  parallel  to  the  chords 
which  AX  bisects,  then  for  each 
value  of  X  as  Ap,  the  two  corres- 
ponding values  of  y,  pm  and  pm',  must  be  equal  with  contrary 
signs.  This  can  not  be  the  case  as  long  as  the  equation  of  the 
curve  contains  any  term  with  the  first  power  of  y.  The  reverse 
is  also  true ;  for  if  the  equation  contain  no  term  with  the  first 
power  cf  y,  for  each  value  of  x  there  will  be  two  equal  values  of  y 
^vith  contrary  signs,  and  these  two  values  taken  together  will  form 
a  chord  bisected  by  the  axis  of  X.  This  axis  will  therefore  be  a 
diameter. 

The  same  reasoning  will  show  that  if  the  axis  of  Y  be  a  di- 
ameter and  the  axis  of  X  parallel  to  the  chords  which  it  bisects, 
the  equation  of  the  curve  can  contain  no  term  with  the  first  power 
of  a;. 


184.     Let  us  now  take  the  general  equation  of  the  second  de- 
gree. Art.  (ICY),  and  see  if  by  any  change  of  the  position  of  the 
axes  of  co-ordinates,  we  can  make  either  of  these  axes  a  diameter. 
For  this  purpose,  let  us  substitute  for  x  and  y,  their  values  taken 
15 


226  INDETERMINATE    GEOMETRY. 

from  formulas  (3),  Art.  (67).     The  new  equation,  leaving  out  the 
dashes  of  the  variables,  will  be  of  the  form, 

^y*   +  V^y  +  **^*   ■\-  qy  •\-  rx   •\-  s  =■   0^ 

in  which 

m  —  {a  tang^  a'  +   6  tang  a'  +  c)  cos»  a' (1). 

?z  =  (a  tang*  a  +  ^  tang  a  +  c)  cos*  a (2). 

-p  =  (2a  tangatanga'+6(tanga4-tanga')  +  2c)cosacosa'...(3), 

q  =  [(2ab'  +  ba'  +  d)  tang  a'  +  (2ca'  +  bb'  +  e)] cos  a'... (4). 

r  =  [(2ab'  +  ba'  +  d)  tang  a  +  (2ca'  +  bb'  +  e)]  cos  a...(o). 

K  now  the  axis  of  X  is  a  diameter,  and  the  axis  of  Y  parallel  to 
the  chords  which  it  bisects,  we  know  from  the  preceding  article, 
that  wc  must  have 

^  =   0,  g  =   0. 

We  have  then  to  assign  such  values  to  the  arbitrary  quantities 
a,  a',  a'  and  &',  as  will  satisfy  the  equations 

2a  tang  a  tang  a'  +  6(tang  a  ■\-  tang  a')  +  2c  =  0 ..(0), 

(2a6'  +  ba'  +  d)  tang  a'  -f  2ca'  +  W   +  e  =  0 (7), 

and  whatever  the  curve  is,  this  can  in  general  be  done ;  for  any 
value  assigned  to  a  in  equation  (6),  taken  with  the  corresponding 
deduced  value  of  a',  will  of  coui-se  satisfy  this  equation.  Tang  a' 
being  thus  fixed,  equation  (7)  can  only  be  satisfied  by  means  of 
values  attributed  to  a'  and  b'.  But  any  value  of  a'  taken  with  the 
■corresponding  deduced  value  of  b'  will  satisfy  this  equation. 

In  the  same  way  it  may  be  shown  that  if  the  axis  of  Y  is  a  di- 
ameter, and  the  axis  of  X  parallel  to  the  chords  which  it  bisects 
we  must  have 

p  =  0,  »•  =  0, 

and  that  these  equations  can  always  be  satisfied- 


INDETERMINATE    GEOMETRY.  227 

If  both  of  the  axes  of  co-ordinates  are  diameters,  at  the  same 
time,  and  each  parallel  to  the  chords  which  the  other  bisects,  wo 
must  have 

^  =   0,  !7  =   0,  r  =  0. 

We  have  seen  above,  that  it  is  always  possible  to  satisfy  the 

equation    p  =  0, (6),     by  assigning  at  pleasure  a  value  to 

either  a  or  a',  and  deducing  the  corresponding  value  of  the  other. 
These  two  angles  being  determined,  a  proper  direction  is  given  to 
the  new  axes  of  co-ordinates,  while  the  new  origin  is  yet  to  be  fixed, 
so  that  we  may  have  at  the  same  time 

<?  =  0,  r  =   0; 

tliat  is 

{^laV  4-  ^«'  +  ^)  tang  a'  +  (2ca'  +  W  +  e)   =   0, 

(2a&'  +  ^a'  +  d)  tang  a  +  (2ca'  +  hV  +  e)   =   0. 

These  equations  being  the  same,  except  that  tang  a  in  one, 
occupies  the  place  of  tang  a!  in  the  other,  it  is  evident  they  can 
not  both  be  satisfied,  at  the  same  time,  unless  we  have  the  terms 
separately  equal  to  0,  that  is, 

2a5'  4-  ha'  +  (Z  =  0,  2m'  +  66'  -f  e  =  0, 

which  ffive  for  a'  and  V  the  values 


&• 


,         1(i£  —  hd  J,         2cd  —  be 

a'  =  — ,  6'   = 

68  —   4ac  6«  —   4ac 

We  recognise  these  values  as  the  co-ordinates  of  the  centre  of 
the  curve,  Art.  (182),  and  therefore  conclude  that  the  new  origin 
must  be  at  the  centre,  and  that  the  new  axes  are  conjugate  di- 
ameters. Art.  (143),  And  since  the  above  values  are  finite  only 
for  the  ellipse  and  hyperbola,  and  infinite  for  the  parabola,  we  con- 
elude  that  both  of  the  co-ordinate  axes  maybe  diameters  at  the  same 
time  in  the  ellipse  and  hyperbola^  but  not  in  the  parabola. 


228  INDETERMINATE    GEOMETRY. 

And  since  there  are  an  infinite  number  of  values  of  a  and  a 
which  will  fulfil  the  above  conditions,  we  conclude  that  m  ike  ellipse 
and  hyperbola,  there  is  an  mfinite  nuniber  of  conjugate  diameters. 

We  have  seen  above  that  equation  jo  =  0,  being  satisfied, 
the  axis  of  X  will  be  a  diameter,  if  we  also  have 

q  =^   Q. 

If  in  this  equation  (7)  we  regard  a'  and  h'  as  variables,  it  will 
be  the  equation  of  a  straight  line,  and  any  values  of  a'  and  h' 
which  are  the  coordinates  of  a  point  on  this  line  will  satisfy  the 
equation,  Art.  (23) ;  hence,  the  new  origin  may  be  any  where  on 
this  line.  But  this  new  origin  must  he  on  the  new  axis  of  X,  and 
may  he  any  where  on  this  axis,  (now  a  dianfieter  of  the  curve). 
Hence 

must  be  the  equation  of  this  new  axis  of  X,  or  diameter,  referred 
to  the  primitive  axes,  a'  and  h'  being  the  variables. 

If  the  axis  of  Y  be  made  a  diameter,  similar  reasoning  will  show 
that    r  =  0     will  be  the  equation  of  this  diameter. 

Tlie  fact  that  g'  =  0  is  the  equation  of  a  diameter,  leads  io 
two  important  conclusions. 

First.  Since  by  assigning  all  possible  values  to  a^  this  equation 
may  be  made  to  represent  all  possible  diameters,  and  since  the  co- 
ordinates of  the  centre,  Art.  (182),  when  substituted  for  a'  and  h\ 
in  this  equation^  must  satisfy  it,  as  they  were  obtained  by  placing 

2a5'  +  Co'  -f  <?  =  0,  2cflr'  -f  5*^  +  e  =   0, 

we  conclude  that  every  diameter  passes  through  the  centre. 

Second.  If  any  straight  line  be  dra\vn  through  the  centre,  and 
the  origin  of  co-ordinates  be  placed  at  the  centre,  and  the  right 
line  be  taken  as  the  axis  of  X,  the  values  of  a'  and  h'  will  satisfy 
the  equation  g'  =  0  ;  and  the  position  of  the  line  being  given, 
a  is  known,  and  the  corresponding  value  of  a',  deduced  fro<n  the 


INDETERMINATE    GEOMETRY.  220 

equation  })  —  0,  will  satisfy  it  also  and  give  a  proper  direction 
to  the  axis  of  Y.  Both  of  these  equations  being  thus  satisfied,  we 
conclude  that  the  right  line  is  a  diameter  ;  hence,  every  right  line 
passing  through  the  centre  is  a  diameter. 


185.  We  have  seen  in  the  preceding  article,  that  both  axes  of 
co-ordinates  can  not  be  diameters  in  the  parabola,  but  that  the  axis 
of  X  will  be  a  diameter  and  the  axis  of  Y  parallel  to  the  chords 
which  it  bisects,  when 

/?  =   0,  »?  =   0, 

and  as  the  equation  when  referred  to  these  axes  is  still  the  equation 
of  the  parabola,  we  must  have.  Art.  (1G9), 

p*  —   4m»  =   0, 

and  since  j9  =  0,  —  4»m  must  equal  0,  But  m  can  not 
be  0,  for  if  it  were,  the  equation  referred  to  the  new  axes  would 
reduce  to 

nx^  -f  ra;  +  «  =  0, 

which  is  the  equation  of  no  curve  ;  hence,  we  must  have  «  =  0, 
and  the  equation  will  reduce  to 

my*  +  rx  -\-  s  =  Q. 

Hence,  in  the  parabola  72  =  0  is  a  condition  consequent 
upon    ^  =   0     and     q  =z   Q. 

This  fact  may  be  verified  thus  :  Since  in  the  parabola  all  di- 
ameters are  parallel,  and  make  with  the  axis  of  X  an  angle  whose 

tangent  is     —  —  ,     Art.  (1*72),  and  since  the  new  axis  of  X  is 
2a 

a  diameter,  we  have 

tanor  a  =    —  — . 
^  2a 


230  INDETERMINATE    GEOMETRY. 

Substituting  this  value  in  equation  (2),  Art.  (184),  we  have 

n      _  «/;2        h^  __        ^*     ,       _         62  —  4ac  _  ^ 

cos^  a        4a "^        2a  4:a  4a 

If  the  axis  of  Y  is  a  diameter,  it  may  be  proved,  in  the  same 
way,  that  we  must  have  m  =  0,  and  that  the  equation  of  the 
parabola  will  take  the  form 

nx*  4-  gry  +  s  =  0. 

It  may  be  further  remarked,  that  any  value  whatever  being  as- 
sumed either  for   tan(/  a  or  tang  aJ  and  substituted  in  equation 

(6),  will,  for  the  parabola,  give     —  —    for  the  value  of  the  other. 

2a 

Also,  if     —  —     be  substituted  in  the  same  equation  for  tang  a 
2a 

or  tang  a',  the  corresponding  value  of  the  other  will  be  - ,  or  in- 
determinate. This  is  evidently  a  consequence  of  the  parallelism 
of  the  diameters  of  the  parabola. 


\r 


/ 


OF    LOCI. 


186.  The  term  locus,  in  Analytical  Geometry  is  applied  to  the 
line  or  surface,  in  which  are  to  be  found  all  of  the  positions 
of  a  po'nt  or  line,  which  changes  its  position  in  accordance  with 
some  determinate  law. 

Thus,  if  a  point  is  moved  in  a  plane,  so  that  it  shall  always  be 
at  the  same  distance  froi^  a  fixed  point,  the  locus  of  the  point  will 
be  the  circumference  of  a  circle. 

Also,  a  plane  tangent  to  a  surface  at  a  given  point,  is  the  locus 
of  all  right  lines  drawn  tangent  to  lines  of  the  surface  at  thia 
point. 


INDETERMINATE    GEOMETRY. 


231 


187.  Tlie  determination  of  the  loci  of  points,  which  are  moved 
in  a  given  plane  subject  to  certain  conditions,  gives  rise  to  a  great 
variety  of  interesting  problems,  several  of  which  it  is  proposed  to 
solve  and  discuss  in  detail,  for  the  purpose  of  indicating  to  the  stu- 
dent the  general  method  to  be  pursued  in  the  solution  of  all. 

It  should  be  remarked,  that  pains  should  be  taken  to  select  the 
best  position  for  the  co-ordinate  axes  in  each  problem,  as  its  solu- 
tion may  be  thus  much  simplified. 

188.  Problem  1st.     To  determine  the  locus  of  a  point,  which 
in  any  of  its  positions  is  at  equal  distances 
from  a  fixed  point  and  fixed  right  line. 

Let  F  be  the  given  point  and  BC  the 
given  right  line.  Through  F  draw  FB  per- 
pendicular to  BC  and  denote  the  known 
distance  FB  by  p.  At  the  middle  point  of 
FB  erect  AY  perpendicular  to  it  and  take 
AX  and  AY  as  the  co-ordinate  axes.  Let 
M  be  any  position  of  the  moving  point,  the  co-ordinates  of  which 
are  AP  =  x,  and  PM  =  y.  By  the  conditions  of  the  prob- 
lem, we  must  have 


But 


MF  =  MC. 


MF 


VMP'    +  FP' 


=  yy' 


+  (.  -  l)\ 


and 


MC  =  BP  =  BA  -f  AP  =  ar   -f-  |» 


henoe 


232  INDETERMINATE    GEOMETRY. 


V 


y^  ^   {x   -  I)    =  X  +1 

Z  2i 


Squaring  both  members  and  reducing,  we  obtain 
y2  =    2^.r, 

an  equation  expressing  the  relation  between  x  and  y  for  all  posi 
tions  of  the  point  M.  It  is  therefore  the  equation  of  the  locus, 
which  is  a  parabola,  Art.  (88). 

189.  Problem  2nd.  To  find  the  locus  of  a  point  moving  in 
such  a  way,  that  the  sum  of  its  distances  from  two  given  points 
shall  always  be  equal  to  a  given  line. 

Let  F  and  F'  be  the  two  given  points,  and  2c  the  distance  be- 
tween them.     Let  2a  represent  the  given  line. 

At  C,  the  middle  point  of  FF',  erect  the  perpendicular  CD  and 
take  CF  and  CD  as  the  co-ordinate  axes. 
Let  M  be  any  position  of  the  point  and 
denote  its  co-ordinates  by  x  and  y,  and 
denote  by  r  and  r'  the  distances  from  the 
point  to  F  and  F'. 

The  right  angled  triangle  FMP  gives 

FM'   =  MP*    +  FP*, 
or,  since     CF  =  c, 

r3  =  2/8  ^   (^  _  cy. 

In  the  same  way  the  right  angled  triangle  F'MP,  gives 

r"  =  y^   ^  [x  ■\-  c)«. 
Adding  these  two  equations,  member  by  member,  we  have 

r2   +  r'2  =   2(y«  +  x^   ■\'  c^) (1), 

tnd  subtracting  them. 


INDETERMINATE    GEOMETRY.  233 

r'«  —  r3  =  4cx,      or      (r'+  r*)  (t-"—   r^)  =  4c2r (2). 

But  by  the  condition  of  the  problem, 

r'   +  r  =   2a (3). 

Substituting  this  in  equation  (2),  we  have 

2cx 
r   ~   r  =  


Combining  this  with  (3),  we  deduce 


,  ex 

r'  =  a   +  _, 
a 


ex 

r  =  a  —  — 

a 


.(4). 


Squaring  these  values  and  substituting  in  (1),  we  obtain 


a^   + 


=  y2   +  a;2   +   c^ 


or 

a8^2  ^  (^a  _  ^2)  ^i  ^   a^  (^a  _   c«), 

or  putting    6*    for     a^  —  c', 

the  same  as  equation  (e).  Art.  (105),  and  the  locus  is  an  ellipse. 


190.  Problem  Sd.  To  find  the  locus  of  any  point  of  a  given- 
right  line,  which  is  moved  so  that  its  extremities  shall  be  con- 
stantly in  two  other  right  lines,  at  right  angles  to  each  other. 

Let  AX  and  AY  be  the  two  right  lines  at  right  angles,  and  M 
any  point  of  the  given  line  CB.     Denote  the  y 

distance  BM  by  a,  and  MC  by  b.  Take  AX 
and  AY  as  the  co-ordinate  axes  and  let 
AP  =  X,      PM  =  ij. 

Since  MP  is  parallel  to  AB,  we  have 

» 

PC    :    MC    ::    AP    :   BM, 


234 

X 

or 

\A 

■^    c 

X. 

whence 

INDETERMINATE    GEOMETRY. 


Vb^  —  y^    :    h    : :    x    :   o ; 


^2.^:2  =  a%'^  —  aV^         or         a^y*   +  h^x*  —  a«6«, 

which  is  evidently  the  equation  of  an  ellipse  whose  semi-transverse 
axis  is  BM  and  semi-conjugate  MC. 


191.  Problem  Uh.  To  find  the  locus  of  the  centres  of  all  cir- 
cles which  pass  through  a  given  point  and  are  tangent  to  a  given 
right  line. 

Let  M  be  the  given  point,  and  BX  the  given  line.     Through  M, 

draw  MA  perpendicular  to  BX,  and 
let  AX  and  AM  be  the  axes  of  co- 
ordinates. Denote  the  ordinate 
MA  by  J9,  the  ab.-cissa  of  this  point 
ii        P       -^  X.  /  will  be  0.     Let  C  be  the  centre  of 

one  of  the  circles  and  denote  its  co-ordinates  by  x'  and  y'.     The 
equation  of  this  circle.  Art.  (34),  will  be 

{x  -  x'Y  -f  (y  -  y'Y  =  R«. 

But  since  it  passes  through  the  point  M,  the  co-ordinates  of  this 
point  will  satisfy  the  equation,  and  give 

^'2    ^    (p   -   y'Y   =   R2, 
and  since  the  circle  is  tangent  to  BX,  we  have     R  =  y' ;     hence 

x'^  +  {p  -  y'Y  =  y'^ 


ar'«  —   '2p7j'  =   —  p\ 


INDETERMINATE    GEOMETRY. 


235 


which  expresses  the  relation  between  x'  and  y'  for  any  position  of 
the  circle,  it  is  therefore  the  equation  of  the  locus. 

If  the  origin  be  now  transferred  to  V  midway  between  M  and 
A  the  formulas  (2)  of  Art.  (67)  become 


aj'  ^-  :r. 


y'  =  y  + 


I> 


the  substitution  of  which  gives 

rca  =    2^9?/, 

the  equation  of  a  parabola  of  which  M  is  the  focus  and  BX  the 
directrix,  and  this  is  evidently  another  method  of  enunciating  and 
solving  problem  1st,  Art.  (188). 


192.  Problem  hih.  To  find  the  locus  of  the  intersection  of 
right  lines,  drawn  from  the  extremities  of  the  transverse  axis  of  a 
given  ellipse,  to  the  extremities  of  chords  of  the  ellipse  perpendic- 
ular to  the  transverse  axis. 

Let  ABD  be  the  given  ellipse  and  DD'  any  chord  perpendicular 
toAB.  Through  D  and  D' 
draw  the  lines  AD  and  BD', 
it  is  required  to  find  the  locus 
of  M,  their  point  of  intersec- 
tion. Let  the  equation  of  the 
given  ellipse  be 


and  denote  the  co-ordinates  of  the  point  D  by  x'  and  y'.     Tlte 
equation  of  condition  that  this  point  shall  be  on  the  ellipse  will  be 


a*y'%  +  fcV8  =  a«6«, 


or 


J2 


y'^  =  -,{<^'  -  ^') (1). 


236  INDETERMINATE    GEOMETRY. 

The  equation  of  the  right  line  AD,  passing  through  the  two 
points  A  and  D,  Art.  (31),  will  be 


=  ^1_(^  +  a) (2), 


and  of  the  line  D'B,  ^  -^— , 

—    v'  I   /  ''' 

y  =  -^-^{x-a) (3).  :t 

x*  —   a  2) 

Multiplying  these  equations,  member  by  member,  we  have 


in  which  y  and  x  are  the  co-ordinates  of  the  point  of  intersection, 
for  the  two  particular  Hues  AD  and  D'B.  If  y'  and  x'  be  elimi- 
nated from  this  equation,  it  is  evident  that  y  and  x  will  belong  to 
no  particular  lines,  but  will  be  the  co-ordinates  of  the  point  of  in- 
tersection of  all  the  lines  w^iich  fulfil  the  required  condition  ;  and 
the  resulting  equation  will  be  the  equation  of  the  required  locus. 
Substituting  the  value  of  y'^  taken  from  equation  (1)  in  equation 
(4),  it  reduces  to 


a 


which  is  the  equation  of  an  hyperbola  having  the  same  axes  as  the 
given  ellipse,  Art.  (105). 

This  method  of  determining  loci,  by  combining  two  equations 
belonging  to  particular  lines,  so  as  to  eliminate  the  arbitrary  con- 
stants which  serve  to  determine  the  position  of  the  lines,  thus  de- 
ducing an  equation  independent  of  these  constants,  and  therefore 
belonging  to  all  lines  which  fulfil  the  required  condition,  is  of  fre- 
quent use. 

193.   Problem  6th.  If  from  the  extremity  of  a  diameter  of  a  circle 


INDETERMINATE    GEOMETRY. 


237 


any  straight  line,  as  AR,  be  draAvn  until  it 
intersects  the  tangent  BR  at  the  other  ex- 
tremity, and  the  distance  AM  be  laid  off 
equal  to  NR,  it  is  required  to  find  the 
locus  of  M.  Let  A  be  the  origin,  and 
AB  and  AY  the  co-ordinate  axes.  Let 
AB  =  2a,  AP  =  a-,  PM  =  y.  Then 
drawing  NP'  parallel  to  MP,  we  have 


AP    :    PM 


AF    :    P'N. 


Also,  since     AM  =  NR,     AP  =  P'B, 

P'N  =  VP'B  X   AP'  =  Vj(2a 
The  above  proportion  then  becomes 

X    :    y   :  :   2a  —  a:  :  ^/x{^a  —  x) ; 
whence 

1/^  =  -: or  y 


x). 


2a 


^  2a  —  X 


for  the  equation  of  the  locus.  The  equation  being  of  the  third 
degree,  the  line  is  of  the  third  order.  Art.  (33). 

All  negative  values  of  x  give  imaginary  values  for  y. 

X  =:  0     gives     y  .=   dz  0. 

Each  positive  valne  of  a;  <  2a  gives  two  real  values  of  ?/, 
equal  with  contrary  signs. 

a:  =   2a     gives     y  ::=    rt:    co . 

All  positive  values  of  a;  >  2a  give  imaginary  values  for  y, 
and  the  curve  is  as  indicated  in  the  figure,  the  line  BR  being  an 
asymptote.  Art.  (161).     It  is  called  the  Cissoid  of  Diodes, 


194.     Tlie   following    ^.Toblems   may   be   solved  by  puisuing 
methods  similar  to  those  indicated  in  the  preceding  articles. 


!38 


[NDETEKMINATE    GEOMETRY. 


7.  To  find  the  locus  of  a  point  moving  in  sucL  a  way,  that  tht 
difference  of  its  distances  from  two  given  points  shall  always  be 
equal  to  a  given  line. 

8.  Given  the  line  AB  and  the  two  lines 
DB  and  AD',  to  find  the  locus  of  M  moving 
so  that  MP  shall  be  a  mean  proportional 
li      between  PC  and  PD.  CLJ^-*^^/ 

9.  Given  the  base  of  a  triangle  and  the  difference  of  the  angles 

at  the  base,  to  find  the  locus  of  the  vertex.      {J,,!U^^'     •^^Lt^.^^jUv^ 

10.  Given  the  base  of  a  triangle,  to  find  the  locus  of  the  vertex 
when  one  anofle  at  the  base  is  double  of  the  other.        '   •  .       --t-' 


11.  To  find  the  locus  of  the  point  of  inter- 
section of  a  tangent  to  an  ellipse,  with  a  per- 
pendicular let  fall  upon  it  from  either  focus. 

12.  Given  the  semi-circle  ASB,  to  find  the 
locus  of  the  point  M,  so  that  we  may  always 
have     *"}''(  J '''.   f^    O^^^r'^sJi^  , 

AP    :    PS    :  :    AB    :    PM. 

13.  Given  the  indefinite  right  line  AB, 
the  point  C,  and  the  perpendicular  CD,  to 
find  the  locus  of  M  so  that  we  may  «ilways 
have     MR  =  AD. 


(i,0^/y^Q)^^J>^  L    &^ 


OP   SURFACES    OP   REVOLUTION. 


195.  A  surface  of  revolution,  is  a  surface  which  may  he  gene- 
rated hy  revolving  a  line  about  a  right  line  as  an  axis. 

By  revolving,  is  to  be  understood,  moving  the  line  in  such  a 
manner,  that  each  point  of  it  will  generate  the  circumference  of  n 


INDETERMINATE    GEOMETUY.  239 

circle  whose  centre  is  in  the  axis,  and  whose  plane  is  perpendicular 
to  the  axis.     The  moving  line  is  called  the  generatrix. 

From  the  definition  it  follows,  that  every  plane  perpendicular 
to  the  axis  will  cut  a  circle  from  the  surface. 

Every  plane  passed  through  the  axis  will  cut  from  the  surface 
a  meridian  curve,  or  line,  and  if  this  be  revolved  about  the  axis  it 
will  generate  the  surface. 


196.  In  order  to  obtain  the  general  equation  of  a  surfoce  of 
revolution,  Art.  (54),  let  us  take  the  axis  of  the  surface  for  the  axis 
of  Z,  and  the  co-ordinate  planes  at  right  angles.  The  general 
equation  of  the  generatrix  will  then  be.  Art.  (52), 

=c  =  /W,  !/  =  /W (1), 

and  let  r  denote  the  distance  of  any  point  of  this  line  from  the  axis. 
Since,  from  the  nature  of  the  surface,  this  point  in  its  revolution 
must  describe  a  circle  whose  centre  is  in  the  axis  of  Z,  and  whose 
plane  is  perpendicular  to  this  axis,  that  is  parallel  to  the  plane  XY, 
we  must  have  in  every  position  of  the  point, 

a:*   -f  y«  =   r^ (2), 

and  since  this  point  is  on  the  generatrix,  the  values  of  x  and  y 
taken  from  equations  (1),  must  fulfil  the  condition  '  expressed  by 
equation  (2),  and  give 

W)'  +W)'  =  »••• 

Equating  these  two  values  of  r^,  we  have 

.r«  +  ,,'  ='filf  +'f\zf (3), 

an  equation  expressing  the  relation  between  the  co-ordinates  of  the 
point  in  all  of  its  positions.  It  is  therefore  the  equation  of  the 
stirface,  in  which  /(z)  and  /'(z),  are  the  values  of  x  and  y  ob- 
tained by  solving  the  equations  of  the  generatrix. 


240  INDETERMINATE    GEOMETRY. 

197.  To  illustrate,  let  us  find  the  equation  of  a  surface  gene- 
rated by  revolving  a  right  line  ahoui  an  axis  not  in  the  same  plane 
with  it. 

The  axis  of  revolution  being  taken  as  the  axis  of  Z,  we  may- 
take  for  the  equations  of  the  generatrix,  Art.  (44), 

X  =  az  +  a,  y  =z  hz  +  ^, 

from  which,  w^e  have 

f(z)  =  az  +  a,  f(z)  =.  bz  +  (3. 

Substituting  these  in  equation  (3),  it  becomes 

x^  +  7/^  =   [az  +  ciy  +   (bz  +  /3)2. 

If  the  axis  of  X  be  assumed  perpendicular  to  the  generatrix  and 
intersecting  it,  the  projection  of  the  generatrix  on  the  plane  XZ 
will  be  parallel  to  the  axis  of  Z,  and  its  projection  on  the  plane  YZ 
will  pass  through  the  origin  of  co-ordinates  ;  hence,  Art.  (45),  we 
have 

a  =  0,  /3  =  0, 

and  the  above  equation  becomes 

a;3  +  y2  _  bH^  =   a2 (1). 

If  we  intersect  this  surface  by  a  plane  parallel  to  XY,  the  equa- 
tion of  which,  Art.  (62),  is 

z  =  Cy  X  and  y     indeterminate, 

we  shall  obtain,  Art.  (62), 

x'i  -\-  y^  =   bh^  +  a', 

for  the  equation  of  the  projection  of  the  intersection  on  the  plane 
XY,  which  represents  a  circle  whose  radius  is  Vb^c^  -\-  a^ 
Art.  (35) ;  and  this  circle  will  be  real,  whatever  be  the  value  of  c  ; 
and  the  smallest  possible  when     c  =  0,     in  which  case  the  cut- 


INDETERMINATE    GEOxMETRY.  241 

(ing  plane  is  the  plane  XY,  Art.  (62).     And  since  this  projecticn 
is  equal  to  the  intereection  itself,  we  sec  that  every  intersection  by 
a  plane  perpendicular  to  the  axis  will  be  a  circle,  as  we  know  ifc 
should  be,  from  the  definition  of  the  surface. 
If  we  make     y  =  0     in  equation  (1),  we  have 


^2^2  =  a2,         or         62^2 


j2   =    „   a«, 


for  the  intersection  by  the  plane  XZ,  Art.  (62). 

If  we  make     x  =  0,     we  have  for  the  intersection  by  the 
plane  YZ, 

hH^  -   2/2  =   -   a«, 

and  these  are  evidently  the  equations  of  two  equal  hyperbolas,  the 
conjugate  axis  of  each  lying  on  the  axis  of  Z,  Art.  (105).  And 
since  the  surface  may  be  generated  by  revolving  either  of  these 
meridian  curves  about  the  axis,  it  is  called  a  hyperholoid  of  revolu- 
tion of  one  nappe.  Of  one  nappe,  since,  as  is  readily  seen,  it  forms 
one  uninterrupted  surface. 


198.  If  the  generatrix  is  in  the  plane  with  the  axis  of  revolution, 
this  plane  may  be  taken  for  the  plane  XZ,  and  as  before,  the  axis 
of  revolution  for  the  axis  of  Z,  in  Avhicli  case  the  equations  of  the 
generatrix  will  be.  Art.  (52), 

^  =  A^\  y  ==  f'{^)  =  0, 

and  equation  (3)  of  Art.  (196)  will  reduce  to 

x^   +  y^  =  fizj' (1), 

in  which  /(::)   is  the  value  of  x  deduced  from  the  equation  of  the 
generatrjc. 


xU' 


Examples. 

1.    ITie  equation  of  a  right  line  in  the  plane  XZ,  and  passiuo- 
16 


242 


INDETERMINATE    GEOMETRY. 


through  a  point  on  the  axis  of  Z,  whose  co-ordinates  are    x*  =  0, 

z'  =  c,     will  be,  Art.  (29),    y  ^y'  -»-  ;i  ♦  y 

from  which  we  have 


'  j\z)  =  a(z  -  c). 

This  substituted  in  equation  (1),  gives 
iC«  -f  y«  =  a«(2  —  c)«, 

for  the  equation  of  the  cone  generated  by  revolving  the  right  line 
about  the  axis  of  Z.     This  equation  may  be  put  under  the  form 

(*«   +  y«)i.  =   (z  -   of, 
a* 

or  denoting  the  angle  made  by  the  generatrix  with  the  base  by  v^ 
we  have 


_  r=  tangv; 
a 


whence 

{x^  +  2/2)  tang«  v  =  (z  -  c)\ 
the  same  equation  as  that  deduced  in  Art.  (80). 

2.  If  the  axis  of  a  parabola  in  the  plane  XZ,  coincide  with  the 
Axis  of  Z,  and  its  vertex  be  at  the  origin  of  co-ordinates,  its  equa- 
tion will  be,  Art.  (84), 

V—  ^  x^  =   2p«, 

from  which  we  have 

f{z)  =  V2fz,  f{^  =   2pz, 

which  substituted  in  equation  (1),  gives 


IKDRTERMINATE    GEOMETUY. 

X*   -\-  y^  =   2pz, 


243 


for  the  equation  of  the  surface  generated  by  revolving  a  parabola 
about  its  axis ;  called  a  paraboloid  of  revolution. 

3.  If  the  transverse  axis  of  an  ellipse,  in  the  plane  XZ,  lies  on 
the  axis  of  Z,  and  its  centre  is  at  the  origin  of  co-ordinates,  its 
equation  will  be,  Art.  (105), 


a«x2   +  i^^a  =  a8j2^ 


whence 


t* 


x^  =  -,(««  -  .»)   =f{z)\ 
and  this  in  equation  (1),  gives 

X*  J^  y"^  =  -  (a*  _  22),      or       a\x'^  +  y^)  +  hH^  =  a«6«...(2), 
a* 

for  the  equation  of  a  surface  generated  by  revolving  an  ellipse 
about  its  transverse  axis. 

If  the  conjugate  axis  of  the  ellipse  lies  on  the  axis  of  Z,  the  equa- 
tion will  be, 

a«2«  -f  h^x^  =  a%^,         whence         ^*  =  ^  (^^^  ~  ^'^)  ^  /=  ' 

and  the  equation  of  the  surface 

h\x^   +   7/)   +  a'^z^  =   a%^ (3). 

These  surfaces  are  called  ellipsoids  of  revolution  ;  or  spheroids. 
The  first  is  tlie  prolate^  and  the  second  the  oblate  spheroid. 

If  in   either  of  equations  (2)  and  (3)  we  make     a  =  6,     the 
ellipse  becomes  a  circle,  and  the  equation  reduces  to 

x'^  -f  y«  +  2«  =   a«, 
for  the  equation  of  a  sphere. 


4.  If  in  equations  (2)  and  (3)  we  change  6*  into  —  &',  we  have 


f44  INDETERMINATE    GEOMETRr, 

and 

Ihe  firet  represents  the  surface  generated  by  revolving  an  hyper- 
bola about  its  transverse  axis,  or  hyperholoid  of  revolution  of  two 
nappes.  Of  two  nappes,  since  it  consists  of  two  distinct  parts,  one 
being  generated  by  one  branch  of  the  hyperbola,  and  the  other  by 
the  other  branch. 

The  second  represents  the  surface  generated  by  revolving  the 
hyperbola  about  its  conjugate  axis.  Its  equation,  after  dividing 
by  ^V^comes 

of  the  same  form  as  equation  (1),  Art.  (197).  From  which  we  see 
that  this  surface  may  not  only  be  generated  by  revolving  an  hyper- 
bola about  its  conjugate  axis,  but  also  by  revolving  a  right  lino 
about  another,  not  in  the  same  plane  with  it. 


OF    SURFACES    OF   THE    SECOND   ORDER. 

199.  Surfaces,  like  lines.  Art.  (33),  are  classed  into  orders  ac- 
cording to  the  degree  of  their  equations. 

"We  have  seen,  Art.  (57),  that  the  plane  is  the  only  surface  of 
the  first  order. 

Tlic  equation  of  every  surface  of  the  second  order  must  be  a 
particular  case  of  the  most  general  equation  of  the  second  degree 
between  three  variables, 

mx^  +  'i^y^  +  P^'^  +  m'xy  -^  n'xz  +  p'yz 

-4- m"ar -f   n"?j    -f  p"z   +  I  =   0 (1), 

which,  for  the  same  reason  as  th^t  given  in  Art.  (167),  may  be 


INDETERMINATE    GEOMETRY.  245 

considered  as  referred  to  a  system  of  co-ordinate  planes  at  right 
angles. 

Points  of  the  surfaces  may  be  determined  as  in  Art.  (oo),  by 
assigning  values  to  x  and  y,  and  deducing  the  corresponding  values 
of  z ;  but  the  nature  of  the  surface  will,  in  general,  be  best  ascer- 
tained by  intei-secting  it  by  planes  and  discussing  the  cur\^s  of  in- 
tersection thus  obtained. 


200.  If  we  combine  the  above  equation,  with  the  equation  of  a 
plane  having  any  position,  Art.  (55),  and  then  refer  the  line  of  in- 
tersection to  co-ordinate  axes  in  its  own  plane,  the  resulting  equa- 
tion will  be  of  the  second  degree.  For  one  of  the  equations  being 
of  the  first,  and  the  other  of  the  second  degree,  the  result  of  their 
combination  will  necessarily  be  of  the  second  degree.  We  there- 
fore conclude,  that  the  line  of  intersection  of  any  surface  of  the 
second  order  by  a  plane,  is  a  line  of  the  second  order ^  or  one  of  the 
conic  sections^  Art.  (IVO). 


201.  In  the  surface  represented  by  the  general  equation  of 
Art.  (199),  conceive  a  system  of  parallel  chords  to  be  drawn.  The 
equations  of  one  of  these  chords  will  be  of  the  form.  Art.  (44), 

X  =  az  J(-   a,  rj  =z   hz  •\-  /3 (1), 

and  these  equations  may  be  made  to  represent  any  chord  of  the 
system,  by  giving  proper  values  to  a  and  /3,  a  and  h  remaining  un- 
changed. If  equations  (1)  be  combined  with  the  general  equation 
(1),  Art.  (199),  and  x  and  y  be  eliminated,  a  result  will  be  ob- 
tained of  the  form 

2;«   +  1%  +   1  =  0, 
r  r 

in  which  the  two  values  of  r  will  be  the  ordinates  of  the  points  in 


246  INDETERMINATE    GEOMETRY. 

M^hicli  the  chord  pierces  the  surface.  If  x\  y'  and  z'  denote  the 
co-ordinates  of  the  middle  point  of  this  chord,  since  z'  will  equal 
the  half  sum  of  the  two  values  of  2,  we  shall  have 

,  _  s 

or  putting  for  s  and  r  their  values,  as  found  by  the  actual  combi- 
nation of  the  equations, 

2'  _  _  «(2y^«  +  m'h  -\-n')  +  ^{2nh  -f  m'a  +p')  +  m"a  +  n"h  +p" 
~  2{ma^  +  nh'^  +  p  -\-  tn'ab  -f-  n'a  -f  p'b) 

8ince  the  point     (a;',  y'^  z')     is  on  the  chord,  we  also  have 

x'   —  at'  +  a,  y'   =   hz'  +  /8. 

If  now  these  three  equations  be  combined,  so  as  to  eliminate  a 
and  /3  ;  x'^  y'  and  z'  will  belong  to  the  middle  point  of  no  particu- 
lar chord,  and  the  resulting  equation  will  therefore  represent  the 
locus  of  the  middle  points  of  all  the  chords  of  the  system,  Art. 
(192). 

Combining  the  equations,  by  substituting  for  a  and  ^,  in  the 
first,  their  values  taken  from  the  last,  we  obtain  after  reduction, 

,  _      (2ma-^m'h  +  n')x'  +  (2r?ft  +  m'a-\- p')y'  ■\-m"a -f-  n"b-\-p' 
~  2jL>  -f-  n'a  -}-  p'b 

which  is   ihe  equation  of  a  plane ^  Art.  (5*7).     We  therefore  con 
elude,  thai  every  system  of  parallel  chords  of  the  surface  may  be 
bisecti  J  by  a  plane. 

In  order  that  this  plane  shall  be  perpendicular  to  the  chorda 
which  it  bisects,  we  must  have  the  two  conditions.  Art.  (59), 

_   2ma  4-   m'b  -f   n'  t    __   2nb  +  m'a  +  p^ 

2p  H-  n'a    +  p'b  2p  +   n'a    -h  p^b ' 

and  tliese  equations  can  always  be  satisfied  by  at  least  one  set  of 
real  \  alues  for  a  and  b  ;  for  if  they  be  combined  and  either  a  or  6 


INDETERMINATE    GEOMETRY.  2.47 

eliminated,  there  will  result  an  equation  of  the  third  degree, 
containing  the  other,  which  must  have  at  least  one  real  root,  and 
may  have  three.  Hence,  in  evenj  surface  of  the  se:.ond  order,  there, 
is  at  least  one  plane  which  is  perpendicular  to  the  system  of  chords 
which  it  bisects. 


^ 


202.  Let  such  plane  be  taken  as  the  co-ordinate  plane  XY, 
the  axis  of  Z  being  perpendicular  to  it,  that  is,  parallel  to  the 
chords.  This  plane  will  intersect  the  surface  in  a  line  of  the  second 
order,  Art.  (200),  the  axis  of  which  may  be  determined  as  in  Art. 
(100)  or  (154).  Let  this  axis  be  taken  as  the  axis  of  X  and  a 
line,  perpendicular  to  it  in  the  plane  XY,  as  the  axis  of  Y,  and 
suppose  the  surface  to  be  referred  to  this  new  system  of  co-ordi- 
nate planes. 

Since  the  plane  XY  bisects  a  system  of  chords  parallel  to  the 
axis  of  Z,  the  equation  of  the  surface  must  be  of  such  a  form,  that 
for  every  value  of  x  and  y,  it  must  give  two  equal  values  of  z  with 
contrary  signs.  It  can  therefore  contain  no  terra  involving  the 
first  power  of  s.  Art.  (183).  We  must  then  have  in  the  general 
equation  of  Art.  (199), 

n'  =   0,  y  =  0,  p"  =  0 (1). 

And  since  the  axis  of  X  bisects  all  chords  in  tlie  plane  XY,  par- 
allel to  the  axis  of  Y,  the  equation  of  the  surface  must  also  be  of 
such  a  form  that  for  all  values  of  x,  (z  being  equal  to  0),  there  must 
be  two  equal  values  of  y  with  contrary  signs.  The  equation  can 
then  contain  no  term  involving  the  firat  power  of  y.  We  must 
therefore  have,  in  addition  to  the  above  equations  (1), 

m'   =  0,  n''  =  0, 

and  the  general  equation  (1),  Art.  (199)  must  reduce  to  the  form 

mx^  +  ny^  4-  pz-   -f  m''x   -\-   I  =   0 (3); 


248  INDETERMINATE    GEOMETRY". 

and  as  the  above  transformations  are  always  possible,  tbis  equation 
may  be  made  to  represent  all  surfaces  of  tbe  second  order  by  as- 
signing proper  values  to  the  constants  which  enter  it. 


203.  To  discuss  the  above  equation  more  fully,  let  us  first 
transfer  the  origin  of  co-ordinates  to  a  point  on  the  axis  of  X,  at  a 
distance  from  the  primitive  origin  represented  by  the  arbitrary 
quantity  a',  the  axes  remaiinng  parallel  to  the  primitive.  ITie 
formulas  of  Art.  (72)  become 

X  =  a'  -\-  x',  y  —  y\  2  =  2'. 

Substituting  in  the  above  equation,  we  obtain 

mx'"^  +  ny'"^  +  'pz'^  +  (2ma'  +  m")  x'  +  ma'^  +  m' V  +  Z  =  0...(1 ). 

Since  a'  is  arbitrary,  we  may  assign  to  it  such  a  value  as  to  mate 

2ma'   +  m"  =  0,  or  a'  =    —  — , 

2m 

in  which  case  the  equation,  after  denoting  the  absolute  term  by  } 
and  omitting  the  dashes  of  the  variables,  reduces  to 

mx^  +  ny^  +  'pz^  +  ^'  =   0 (2). 

If  m  =  0,  this  transformation  will  in  general  be  impossible, 
as  we  shall  then  have 


a'  =   —  —  =    00 (3). 

0 


In  this  case  we  may  assign  to  a!  such  a  value  as  will  make 

m"a^   4-   ?  =0,  or  a'  =   —  — , 

m" 

and  equation  (1)  will  reduce  to 

ny«  +  j»z«  •!-  m"x  =  0 (4). 


INDETERMINATE    GEOMETRY.  ^49 

If,  however,  we  have  at  the  same  time  m"  =  0,  this  trans- 
formation will  be  impossible.  But  in  this  case,  equation  (1)  will  at 
once  reduce  to 

ny^  +  V^^   +   Z  =   0,  X  indeterminate (o), 

which  is  evidently  the  equation  of  a  right  cyhnder  with  an  ellipti- 
cal or  hyperbolic  base,  according  as  n  and  p  have  the  same  or  con- 
trary signs,  Art.  (IVO),  the  axis  of  the  cyhnder  coinciding  with  the 
axis  of  X.     Moreover,  in  this  case  equation  (3)  gives 

a'  =  —     indeterminate. 
0 

and  any  point  of  the  axis  of  X  will  fulfil  the  required  condition. 
If    m  =  0,     w  =  0,     equation  (3),  Art.  (202),  reduces  to 

pz^  -f  ^''x  +  ^  =  0,  y   indeterminate. 

If    m  =  0,    p  =.  0,     it  reduces  to 

ny^  +  m"x  -f-  Z  =   0,  z   indeterminate ; 

both  of  which  are  equations  of  right  cylinders  with  parabolic  bases, 
the  elements  of  the  first  being  parallel  to  the  axis  of  Y,  and  those 
of  the  second  parallel  to  the  axis  of  Z,  Art.  (V6). 

If    m"  =   0     also,  in  the  last  two  equations,  the  first  will  give 


2  =   ±  \/  —  _  ,  X  and  y  indeterminate^ 

V  p 

which  represents  two  planes  parallel  to  the  plane  XY,  Art.  (C2); 
which  are  real  when  I  and  p  have  contrary  signs  ;  become  one 
when  Z  =  0  ;  and  are  imaginary  when  I  and  p  have  the  same 
sign  ;  and  are  particular  cases  of  the  cylinder^  analogous  to  the 
particular  cases  of  the  parabola  discussed  in  Art.  (I7l). 

In  the  same  way  it  may  be  proved,  that  the  second  equatioii 
will  represent  two  planes  parallel  to  the  plane  XZ. 


260  INDETERMINATE    GEOMETRY. 

If  7?i  =  0,  ?^  r=  0,  i>  =  0,  the  equation  ceases  to  be 
one  of  the  second  degree. 

From  this  discussion,  we  see  that  all  surfaces  of  the  second  order 
vnW  belong  to  one  of  the  three  classes  represented  by  the  following 
equations. 


First, 

mx^  -f  ny"  +  pz^  -{-  I 

=   0. 

Second, 

ny^  +  pz^    +  m"x  —  0. 

ny^  +  pz^   +  I  =  0'' 

TJiird, 

pz^  +  m"x  +  I  z=  0 
ny^  +  'ni"x  +1  =  0^ 

>•  . 

204.  The  centre  of  a  surface  is  a  point,  through  which  if  any 
straight  line  be  drawn  terminating  in  the  surface,  it  will  be  bisected 
at  this  point. 

If  the  origin  of  co-ordinates  be  placed  at  the  centre,  it  is  erident 
that  for  every  point  on  one  side  of  this  origin,  there  must  also  be 
another  in  a  directly  opposite  direction,  at  the  same  distance,  and 
having  the  same  co-ordinates  with  a  contrary  sign.  Hence,  the 
equation  of  the  surface  must  be  of  such  a  form,  that  it  will  not 
change,  when  for  +  a?,  +  y  and  +  z,  —  ^,  —  y 
and  —  z  are  substituted ;  that  is,  all  of  its  terms  must  be  of  an 
even  degree  with  respect  to  the  variables. 

In  order  then  to  ascertain  whether  a  given  surface  has  a  centre  ; 
we  see  if  all  the  terms  of  its  equation  are  of  an  even  degree,  if  so, 
the  origin  of  co-ordinates  is  a  centre  ;  if  they  are  not,  we  then  see 
if  the  origin  of  co-ordinates  can  be  so  placed  as  to  make  all  the 
terms  of  the  transformed  equation,  of  an  even  degree.  If  this  is 
possible,  the  surface  will  have  a  centre,  which  will  be  at  the  new 
origin.     If  it  is  not  possible,  the  surface  will  have  no  centre. 

205,  By  applying  the  above  principles  to  surffices  of  the  second 


INDETERMINATE    GEOMETRY.  251 

order,  >ve  see  that  all  of  the  first  class  have  centres.  That  none  of 
the  second  have  centres.  That  the  cyhnders  represented  by  the 
first  equation  of  the  third  class  have  an  infinite  number  of  centres, 
each  point  of  the  axis  fulfilling  the  required  condition.  That  those 
represented  by  the  second  and  third  equations  have  no  centres. 


206.  Any  plane  which  bisects  a  system  of  parallel,  chords  of  a 
surface,  is  called  a  diametral  plane  ;  and  if  the  chords  are  perpen- 
dicular to  the  plane,  it  is  a  principal  diametral  plane,  or  simply  a 
principal  plane. 

Two  diametral  planes  intersect  in  a  diameter  common  to  the  two 
curves  cut  from  the  surface  by  these  planes,  and  this  intersection 
is  also  a  diameter  of  the  surface  ;  and  two  principal  planes  intersect 
in  an  axis  of  the  surface. 

A  diametral  plane  may  be  constructed,  by  drawing  three  par- 
allel chords  of  the  surface,  not  in  the  same  plane,  and  bisecting 
them  by  a  plane.  By  constructing  two  planes  in  this  way,  we  de- 
termine a  diameter,  and  the  middle  point  of  this  diameter  will 
evidently  be  the  centre. 


207.  The  co-ordinate  planes  being  at  right  angles  to  each  other, 
we  see  that  each  of  them,  in  surfaces  of  the  second  order  of  the 
first  class,  is  a  principal  plane.  For,  if  equation  (2),  Art.  (203),  be 
solved  with  reference  to  either  variable,  we  shall  have  two  equal 
values  with  contrary  signs,  and  these  two  values  taken  together, 
will  form  a  chord,  perpendicular  to  the  co-ordinate  plane  of  the 
other  two  variables,  and  bisected  by  it. 

From  this,  it  also  follows  that  the  axes  of  co-ordinates  are  axes 
of  the  surface.  Art.  (206). 

In  the  second  class,  equation  (4),  Art.  (203),  the  co-ordinate 
planes  ZX  and  YX,  are  also  principal  planes,  and  the  axis  of  X  is 
an  axis  of  the  surface. 


252  INDETERMINATE    GEOMETRY. 

In  the  cylinders  represented  by  the  first  equation  of  the  third 
class,  the  planes  ZX  and  YX  are  principal  planes,  and  the  axis  of 
X  is  the  axis  of  the  cylinder. 

In  the  cylinders  represented  by  the  second,  the  plane  XY  is  the 
only  principal  plane,  and  there  is  no  axis. 

^     In  those  represented  by  the  third,  the  plane  ZX  is  the  only 
.A\j)rincipal  plane,  and  there  is  no  axis. 


DISCUSSION    OF    THE    VARIETIES    OF     SURFACES    OF    THE 
SECOND    ORDER. 

208.  All  the  varieties  of  the  first  class  of  surfaces  of  the  second 
order,  or  those  which  have  a  single  centre,  may  be  obtained  by 
making  in  their  equation,  Art.  (203). 

First,     w,  n  and  p  all  positive,  I  being  negative  or  positive. 

Second.  Either  two  positive  and  the  other  negative,  I  being 
positive. 

Third.  One  positive  and  the  other  two  negative,  I  being  posi- 
tive. 

For  if  all  are  negative,  the  signs  of  both  members  of  the  equa- 
tion may  be  changed,  giving  the  first  case. 

If  two  are  negative,  the  other  positive  and  I  negative,  the  signs 
may  be  changed,  giving  the  second  case. 

If  one  is  negative,  the  others  positive,  and  I  negative,  the  signs 
may  be  changed,  giving  the  third  case. 

First,  in,  n  and  p  positive,  I  negative  or  positive. 

209.  Supposing  I  to  be  negative,  the  equation  of  the  first  class, 
Art.  (203),  may  be  put  under  the  form 

mx^  +  717/^  -f  pz^  =  I (1). 

Let  us  intersect  this  surface  by  planes  parallel,  respectively,  t^ 


INDETERMINATE    GEOMETRY.  263 

the  co-ordinate  planes  ZY,  ZX  and  XY.  The  equations  of  the 
cutting  planes,  Art.  (62),  will  be 

X  —  h,  y  =  k,  z  =  9. 

Combining  these  with  the  equation  of  the  surface,  Art.  (62),  we 
obtain 

mx^  ■\-  2^^^  =   I  —    nk"^', (2), 

for  the  equations  of  the  projections  of  the  several  intersections  on 
the  co-ordinate  planes  ;  and  since  the  curves  are  parallel  to  the 
]ilanes  on  which  they  are  projected,  the  projections  are  equal  to 
the  curves  themselves. 

Each  of  these  equations  represents  an  elhpse,  Art.  (169),  and 
these  ellipses  will  be  real  when  the  second  members  of  the  equa- 
tions are  positive,  or 


h  <  ±\/l,       k  <  ±\/L,       a  <  ±\/i. 

If 

h=.±:JL,  l-=±:Jl,  ff=dz\/L, 

the  above  equations  reduce  to 

?iy'  -f  pz^  =  0,      mx^  -f-  pz^  =  0,      mx'^  -f  ny"^  =  0, 

and  the  first  members  of  each  being  the  sum  of  two  positive  quan- 
tities, they  can  only  be  satisfied  by  making 

y=:0,     z=0;       ar  =  0,     z  =  0;       a:  =  0,     y=.0, 

which  are  the  equations  of  points 


2:>4 


INDETERMINATE    GEOMETRY. 


If 


>±\fL,        k>±^/l,        Jy±^Jl, 


the  second  members  of  the  above  equations  (2)  will  be  negative. 
and  they  can  be  satisfied  by  no  values  of  the  variables,  and  the 
ellipses  will  be  imaginary,  that  is,  the  planes  will  not  intersect  the 
surface. 
If 


;t  =  0, 


/7  =   0, 


equations  (2),  become 


pz^  =  I, 


mx^  +  ^^y^ 


^, 


which  are  the  equations  of  the  principal  sections,  and  each  of  these 
sections  is  evidently  larger  than  any  other  made  by  a  parallel 
plane. 

From  this  discussion  we  conclude  that  if  the  surface,  represented 
by  equation  (1),  be  intersected  by  a 
system  of  planes  parallel,  respectively, 
to  the  co-ordinate  planes,  the  curves 
of  intersection  will  all  he  ellipses^  and 
these  ellipses  will  diminish  as  the  dis- 
tance of  the  cutting  plane  from  the 
centre,  on  either  side,  is  increased,  un- 
til they  reduce  to  points ;  after  which  there  will  be  no  intersection 
and  no  points  of  the  surface.  The  surfjace  is  then  limited  in  all 
directions,  as  in  the  figure,  and  is  called  an  UUijJsoid. 

Ifw^emake     y  =   0,     z  =  0      in  equation  (1),  we  have 


mz'^  =  /, 


or 


CB  or  CA. 


In  A  similar  way  we  find 


INDETERMINATE    GKU.VlElllY.  265 

JL  =  CE  or  CE',  =t\/-  =  CD  or  CD'. 


Placing  the  expressions  for  these  semi-axes,  respectively  equa' 
to  a,  h  and  c,  we  have 

^    m  ^  n  ^  p 

whence 

and  substituting  these  in  equation  (1),  we  obtain 

•^   4-  2^    4-    !!.  =   1 
or 

an  equation  for  the  ellipsoid,  referred  to  its  centre  and  axes,  analo- 
gous to  equation  (e),  Art.  (105). 

'^        210.     If    m  =  w,     equation  (1)  of  the  preceding  article  may 
be  put  under  the  form 

..  +  ,«  =  LZU!!!  =  7(i)^ 

771 

which  is  the  equation  of  a  surface  of  revolution,  the  axis  of  Z  being 
the  axis  of  revolution.  Art.  (198).  But  since  m  =  w,  we  have 
a  =  b  or  CA  =  CE,  and  the  surface  is  generated  by  re- 
volving the  ellipse  "PDA  about  its  conjugate  axis,  and  is  the  oblate 
$pheroid,  Art.  (198). 

Likewise  if    7i  =  p,     equation  (1),  becomes 

„    .      „         I  —  mx^         TTx* 
y«   +  z^  = =  f{x) 


256  INDETERMINATE    GEOMETRY. 

which  is  the  equation  of  the  prolate  spheroid. 
If    m  =^  n  =^  p,     we  obtain 

X''    +   y^    -\-   Z'   :=   — , 
m 

which  is  the  equation  of  the  sphere^  Art.  (198). 
If    Z  =  0,     equation  (1)  becomes 

mx^  +  mj^  +  pz^  =   0, 

which,  since  the  first  member  is  the  sum  of  three  positive  quanti- 
ties, can  only  be  satisfied  by  making 

^  =   0,  3/  =   0,  2=0, 

which  are  the  equations  of  a  pointy  Art.  (41). 

If  I  is  positive,  equation  (2),  Art.  (203),  takes  the  form 

mx^  4-  nif  +  pz"^  z=z   —   l^ 

which  can  be  satisfied  for  no  values  of  x^  y  and  z,  and  therefore  re- 
presents no  surface,  or  an  imaginary  surface. 

From  this  discussion  we  see  that  the  particular  cases  of  the  El- 
lipsoid are,  the  ellipsoid  of  revolution,  the  sphere,  the  point,  and 
the  imaginary  surface. 


Second^  m  and  n  positive^  p>  negative  and  I  positive. 

211.     In  this  case  equation  (2),  Art.  (203),  takes  the  form 

mx^  +  mf  -^  pz"^  =   —  I (1). 

Intersecting  the  surface  by  planes  as  in  Art.  (209),  we  have,  for 
the  equations  of  the  projections  of  the  curves  of  intersection, 

ny"^    —  pz^  =   —  I  —  mh^\ 

mx^  —  pz'^  =   —   I   -   nJc^ ; (2) 

mx^  -if  ny"^  =   —   I  +  pg\ 

Rach  of  thi-  first  two  of  these  equations  represents  an  hyperbo- 


INDETERMINATE    GEOMETRY. 


257 


la,  whose  transverse  axis  coincides  with  the  axis  of  Z,  Art.  (105). 
and  which  increases  in  length,  indefinitely  a&  h  and  k  increase. 

The  third  equation  rej^resents  an  ellipse,  Art.  (105),  which  i? 
real  when 


P9''  >  I, 


or 


9>   ± 


v/^. 


and  which  increases  as  g  increases, 
when 


This  ellipse  becomes  a  point 


vr 


I, 


V  p 


and  imaginary,  or  there  is  no  curve,  when 


pf-  <  I, 


•(3), 


.   <    ±\/i. 


If      A  =  0,      k  =  0,      g  =  0,      Art.  (62),  equations  (2), 
become 


mj' 


pz^ 


-  I 


mx^ 


J)Z^  —    —   l^  mX^  4-    727/3  =    —   i^ 


which  are  the  equations  of  the  principal  sections. 

The  first  two  represent  hyperbolas,  whose  transverse  axes  are  less 
than  those  of  any  of  the  parallel  hyperbolas.  The  third  equation  can 
be  satisfied  by  no  values  of  x  and  y, 
from  which  it  appears  that  the  plane 
XY  does  not  intersect  the  surface. 

From  this  discussion,  we  conclude, 
that  if  the  surface  represented  by 
equation  (1)  be  intersected  by  a  sys- 
tem of  planes,  parallel  respectively  to 
the  co-ordinate  planes,  the  sections 
parallel  to  ZX  and  ZY,  will  be  hy- 
perbolas having  their  transverse  axes 
parallel  to  the  axis  of  Z,  while   the 

sections  parallel  to  XY,  will  be  ellip- 
17 


258  INDETERMINATE    GEOMETRY. 

ses  when  at  a  greater  distance  from  the  origin,  above  or  below, 
than  the  value  of  ^  in  equation  (3).  Hence,  the  surface  extends 
to  infinity  in  all  directions  from  the  centre,  and  consists  of  two 
distinct  and  equal  parts  or  nappes,  as  in  the  figure.  It  is  therefore 
called,  an  Hyperholoid  of  hoo  nappes. 

If  we  make    y  =  0,    «  =  0,     in  equation  (1),  we  have 


▼  /111 


which  is  imaginary,  and  the  surface  does  not  cut  the  axis  of  X. 
In  a  similar  way,  we  find 

ny^  =   -  Z,  y  =  ±\/EI, 

*      n 

and 

m^  =  I  z   =  ±\/L  =  CA  or  CB. 

V  p 

Placing 


^     m               '        -'       \     n 

we  have 

\ 

"^  =  -?'            **  =   J?* 

-i. 

and  these  in  equation  (1),  give 

X^            7/«            *« 

c«          h^          a? 

1, 

ct 

a%*x^  +  a^Y  -   ^^c^^*  = 

-  a^m 

INDETERMINATE    GEOMETRY.  259 

for  the  equation  of  the  hyperboloicl  of  two  nappes,  referred  to  its 
centre  and  axes. 


212.     If    rri  =  n,     equation  (1)  of  the  preceding  article  may- 
be put  under  the  form 

.«  +  y.  =  -  Lzlf.  =  fi7)\ 

m 

which  is  the  equation  of  a  surface  of  revolution,  Art.  (198),  evi- 
dently generated  by  revolving  the  hyperbola  abouK  its  transverse 
axis  BA,  or  the  hyperholoid  of  revolution  of  two  nappes. 
Tf    /  =   0,     equation  (1)  reduces  to 

mx^  +  ny"^  —  pz^  —  0. 

If  this  surface  be  intersected  by  any  plane  parallel  to  XY,  we 
have  for  the  projection  of  the  intersection 

which  is  the  equation  of  an  ellipse  always  real,  whether  g  be  posi- 
tive or  negative.     If    g  =  0,    we  have 

mx^  -f  ny^  =  0, 

which  can  only  be  satisfied  by 

y  =  0,  X  =  0, 

which  are  the  equations  of  a  point.  If  we  make  first  a:  =  0, 
and  then  y  =  0,  we  obtain  for  the  intersections  by  the  co-or- 
dinate planes  YZ  and  XZ,  the  equations 

nys  —  pz^  =  0,  mx^  —  pz^  =   0, 

or 

y  =  ±:  zsjl,  X  ^  dt  zyl 

^   n  ^  m 

each  of  which  evidently  represents  two  right  lines  passing  through . 


260  INDETERMINATE    QEOMETRf. 

the  origin,  Art.  (109),  and  tlie  surface  can  only  be  a  cone  havimj 
its  vertex  at  the  origin. 

The  j^articular  cases  of  the  Hyperboloid  of  two  nappes  are,  there- 
fore, the  hyperboloid  of  revolution  of  two  nappes^  and  the  cone. 


Third,  m  positive,  n  and  p  negative,  I  positive. 
213.     In  this  case,  equation  (2),  Art  (203),  takes  the  form 

mx*  —  wy*  —  pz*  =   —  I (1). 

Intersecting  by  planes,  as  in  Art.  (209),  we  obtain 

ny^  -r  p^^  =  I  -\~  mh^. 

mx^  —  pz-  =   —  I  +  wF (2). 

mx^  —  ny^  =z   —  I  -\-  pg^. 

The  first  of  these  equations  represents  an  ellipse,  which  is 
always  real,  and  increases  as  h  increases  in  either  direction,  from 
the  origin. 

The  second  represents  an  hyperbola,  whose  transverse  axis  coin- 
cides with  the  axis  of  Z  when  the  second  member  is  negative,  or 

wF  <  I,  and  Jc  <   dt\/L, 

and  with  the  axis  of  X,  when 

k   >    ±:sA, 
^  n 

The  third  is  also  the  equation  of  an  hyperbola,  whose  transvers* 
axis  coincides  with  the  axis  of  Y,  when 


P 


v^. 


and  with  the  axis  of  X,  when 


INDETERMINATE    GEOMETRY. 


261 


If  in  the  last  two  of  equations  (2),  we  make 

^  n  ^  p 


we  have 


mx^  —  pz"^ 


0, 


mx^  —  ny^  =  0, 


rr  =   =b 


'      an 


each  of  which  represents  two  right  lines. 


If    A  =  0,    ;& 


0,    g  =  0,     equations  (2)  become 


ny*  +  'P^'^  =  ^,         wi.r*  —  pz^  =  —  Z, 


.2  


mf  =  -  ?, 


for  the  equations  of  the  principal  sections. 

The  first  represents  an  ellipse,  which  is  smaller  than  any  par- 
allel section,  and  is  called  the  ellipse  of  the  gorge.  The  other  two 
represent  hyperbolas.  We  therefore  conclude  that,  if  the  surface 
be  intersected  by  planes  parallel 
respectively  to  the  co-ordinate 
planes,  the  sections  parallel  to  ZX 
and  YX  are  hyperbolas;  while 
those  parallel  to  YZ  are  ellipses, 
always  real,  whatever  be  their  dis- 
tances on  either  side  of  the  centre. 
The  surface  then  extends  to  infinity 
in  all  directions  from  the  centre, 
without  being  separated  into  two 
parts.     It  is  called  an  hyperholoid  of  one  nappe. 

K  we  make     y  =  0,     2=0,     in  equation  (1),  we  have 


262  INDETERMINATE    GEOMETRY. 

^     m 

which  is  imaginary.     In  a  similar  way,  we  find 

CD  =\/i,  CA  =\/l, 

both  of  which  are  real.     Placing 

^      m  ^   n  ^  p 

we  deduce 

I  I  I 

c*  0*  a* 

and  these  in  equation  (1),  give 

for  the  equation  of  the  hyperboloid  of  one  nappe,  referred  to  its 
centre  and  axes. 


214.     If  n  =  ^,     equation  (1)  of  the  preceding  article  may 
be  written 

y.   +  z«  =  L+^  =  f(7)\ 
n 

which  is  the  equation  of  a  surface  of  revolution,  Art.  (198),  evi- 
dently generated  by  revolving  the  hyperbola  about  its  conjugate 
axis,  or  the  hyperboloid  of  revolution  of  one  nappe. 
If    /  =  0,     equation  (1)  reduces  to 

mx^   —  ny^  —  pz^  =  0, 

which  may  be  shown,  as  in  Art.  (212),  to  be  the  equation  of  a 
cone  h;iving  its  vertex  at  the  origin. 


INDETERMINATE    GEOMETRY.  263 

The  particular  cases  of  the  Hyperboloid  of  one  nappe  are,  there- 
fore, the  Jujperholoid  of  revolution  of  one  nappe^  and  the  cone. 


215.  All  the  varieties  of  the  second  class  of  surfaces  of  the 
second  order,  or  those  which  have  no  centre,  may  be  obtained  by 
making  in  equation  (4),  Art.  (203) : 

Mrst.  n  and  p  positive,  m"  being  positive  or  negative  : 

Second,  n  positive  and  p  negative,  m"  being  positive  or  nega- 
tive. 

For,  if  n  and  p  are  negative,  the  signs  of  both  members  of  the 
equation  may  be  changed  giving  the  first  case. 

If  n  is  negative  and  p  positive,  the  signs  may  be  changed 
giving  the  second  case. 


^ 


First,  n  and  p  positive,  m"  positive  or  negative. 


216.  If  m"  is  negative,  equation  (4),  Art.  (203),  may  be  put 
under  the  form 

ny^  +  J52;'  =  'm"x (1). 

Intersecting  the  surface  as  in  Art.  (209),  we  have  for  the  pro- 
jections of  the  several  curves  on  the  co-ordinate  planes, 

ay"*  +  pz^  =  m"h,        pz^  =  ')n"x  —nJc^,        ny^  =  m"x  —  pg^. 

The  first  represents  an  ellipse,  which  is  real  as  long  as  A  is  pos- 
itive, and  increases  indefinitely  as  A  is  increased,  becomes  a  point 
when     A,  =  0,     and  is  imaginary  for  all  negative  values  of  h. 

The  other  two  represent  parabolas,  the  axes  of  which  coincide 
with  the  axis  of  X,  Art.  (84).     And  since  the  parameters  of  these 

parabolas  are,  respectively,    —    and     — ,     whatever  be  the 

J)  n 


264 


INDETERMINATE    GEOMETRY. 


values  of  k  and  ^,  it  follows  that  all  the  parallel  sections  are  equal 
to  each  other. 

By  making     A  =   0,      ;[:  =   0,    ^  =  0,     we  have  for  the 
principal  sections 


ny^   +  P^""  =  0, 


pz^  =  m"x, 


Ihe  first  represents  a  point,  the  origin  of  co-ordinates,  and  each 
of  the  others  a  parabola,  having  its  vertex  at  the  origin. 

From  this  it  appears  that  the  surface  extends  to  infinity  in  the 

positive  direction  of  the  axis  of  X, 
but  does  not  extend  at  all  to  the 
left  of  the  origin ;  that  the  inter- 
sections by  one  system  of  planes  are 
ellipses,  and  by  the  other  two,  para- 
bolas. It  is  therefore  called  an 
elliptical  paraboloid. 


If  m"  is  positive,  equation  (4),  Art.  (203),  takes  the  form 
ny*  -h  P^^  =   —  m"x, 

in  which,  if  we  change  x  into  —  x,  we  shall  have  equation  (1). 
But  the  only  effect  of  this  change  is  to  estimate  the  abscissas  from 
A  to  the  left.  The  equation  will  then  represent  the  same  surface 
revolved  1 80°  about  the  axis  of  Y. 


21*7.     If    n  =  p,     equation  (1)  of  the  preceding  article  may 
be  written 


y3  +   s8  =  — X  =  /r 


which  is  the  equation  of  a  paraboloid  of  revolution^  generated  by 
revolving  the  parabola  about  its  axis,  and  this  is  the  only  particular 
case  of  the  elliptical  paraboloid. 


INDETERMINATE    GEOMETRY.  265 

Second,  n  positive  and  p  negative,  m"  positive  or  negative. 

218.  It  will  only  be  necessary  to  discuss  the  case  where  m"  is 
negative ;  for,  if  m"  is  positive,  it  may  be  shown,  as  in  Art.  (2 1 6), 
that  the  equation  will  represent 'the  same  surface  revolved  180° 
about  the  axis  of  Y. 

This  being  the  case,  equation  (4),  Art.  (203),  takes  the  form 

mf  —  pz^  =  m"x (1). 

Intersecting  the  surface,  as  in  Art.  (216),  we  have 
ny^  —  pz^  =  w"A...(2),     ^?x*  =  —  m"x  -j-  ?iP,     ny^  =  m"x  -[■  pg\ 

The  first  is  the  equation  of  an  hyperbola  always  real,  and  having 
its  transverse  axis  on  the  axis  of  Y  when  h  is  positive,  and  on  the 
axis  of  Z  when  h  is  negative.  Art.  (105).  The  other  two  are  the 
equations  of  parabolas,  the  first  extending  indefinitely  in  the  di- 
rection of  the  negative  abscissas,  and  the  second  in  the  direction 
of  the  positive  abscissas,  Art.  (171). 

By  making  ^  =  0,  ^  =  0,  g  =  0,  we  have  for  the 
principal  sections 

»y*  —  pz^  =  0 (3),  pz^  =  --  m"Xy  ny^  =  m"x. 

The  first  may  be  put  under  the  form 


ny^  ==  pz'',  or  y  =   do  z 


^/i. 


which  represents  two  right  lines  passing  through  the  origin.  The 
other  two  represent  parabolas  each  equal  to  those  cut  out  by  the 
corresponding  parallel  planes. 

From  this,  it  appears  that  the  surface  is  unlimited  in  dii  direc- 
tions ;  that  the  sections  by  one  system  of  planes  are  hyperbolas, 
and  by  the  other  two,  parabolas.  It  is  therefore  called  a  hyperbolic 
paraboloid. 

It  has  no  particular  case. 


ZGQ 


INDETERMINATE    GEOMETRY. 


We  h.'ive  seen  above  that  the  plane  YZ  intersects  the  surface 
in  two  right  lines  represented  by 
equation  (3),  and  that  any  plane  par- 
allel to  YZ,  intersects  the  surface 
in  an  hyperbola,  the  projection  of 
which  is  represented  by  equation 
(2).  If  we  denote  the  ordinate  of 
any  point  of  one  of  these  right  lines 
^y   y'l   to   distinguish  it  from   the 

ordinate  of  a  point  of  the  curve  corresponding  to  the  same  value 

of  Zj  we  shall  have 

Wy'2    _  pz^    =    0. 

Subtracting  this  equation,  member  by  member,  from  equation 
(2),  we  have 


til/ 


i       __       „y/2 


m"h 


whence 


y  —  y 


m"h 


n{y  +  y') 


Now  as  «  is  increased,  y  and  y'  are  both  increased,  and  y  —  y' 
becomes  smaller  and  smaller,  and  when  y  and  y'  become  infinite, 
y  —  y'  becomes  0,  or  the  two  points  coincide  ;  that  is,  the  right 
line  continually  approaches  the  curve  and  touches  it  at  an  infinite 
distance,  or  is  an  asymptote,  Art.  (161).  Hence,  the  two  right  lines 
represented  by  equation  (3),  will  be  the  asymptotes  of  the  pro- 
jections of  the  hyperbolas  cut  out  by  the  planes  parallel  to  YZ. 
Or,  if  two  planes  be  passed  through  these  hues  and  the  axis  of  X, 
the  plane  which  cuts  from  the  surface  an  hyperbola,  will  cut  from 
these  planes,  lines  which  will  be  the  asymptotes  of  the  hyperbola. 


Of   THE    INTERSECTION    OF    SURFACES    OF    THE    SECOND 
ORDER    BY    PLANES. 


219.     It  has  been  proved,  Art.  (200),  that  every  intersection  of 
a  surface  of  the  second  order,  by  a  plane,  is  a  line  of  the  second 


INDETERMINATE    GEOMETRY. 


207 


order.  The  discussion  of  the  nature  of  these  sections,  except  when 
they  are  parallel  to  one  of  the  co-ordinate  planes,  is  much  simpli- 
fied by  referring  them  to  axes  at  right  angles,  in  their  own  planes. 
For  the  purpose  of  this  discussion,  let  us  resume  the  general 
equation,  Art.  (202), 


mx^  -f  ny^  +  pz^  +  m"x  +  r=  0. 


■(1), 


in  which  the  origin  is  at  some  point  A,  on  the  line  AX,  this  being 
the  intersection  of 
two  principal  planes. 
Art.  (206).  Let  any 
plane  be  passed  in- 
tersecting the  surface, 
and  let  A'X'  be  its 
trace  on  the  plane 
XY,  making  an  angle 
/3  with  the  axis  of  X, 
and  let  ^  denote  the 
angle  made  by  this 
plane  with  the  plane 
XY. 

For  any  point  of  the  curve  of  intersection,  as  M,  we  shall  then 
have 


AP', 


y 


PF, 


z  =  MP. 


Let  this  point  be  now  referred  to  the  two  axes  A'X'  and  A'Y', 
at  right  angles  to  each  other  and  in  the  plane  of  the  curve. 
Through  P  draw  PN  perpendicular  to  A'X',  and  PO  parallel  to 
AX  ;  also  draw  NS  perpendicular  to  AX.  Join  M  and  N,  then 
the  angle     MNP  =  &.     Denote  the  distances 


AA'  by  a',         A'N  by  a:', 
ftnd  we  shall  have 


MN  by  y\ 


X  =  a'  +  A'^  +  OP, 


=  NS  -  NO. 


/ 


^  268  INDETERMINATE    GEOMETRY. 

^'    The  right  angled  triangles  MPN,  A'SN,  and  PON,  give 

z  =  y'  sin  ^,  NP  =  y'  cos  ^,  A'S  =  x'  cos  ft 

NS  =  x'  sin  iS,  NO  =  NP  cos  ft        PO  =  NP  sin  /8. 

Substituting  these  values  in  the  preceding  equations,  we  obtain 

z  =  a'  +  X'  cos  ^  -\-  y'  cos  6  sin  /3,       y  =:  x'  sin  ^  —  y'  cos  &  cos  /3. 

If  these  values,  with  the  value,  z  =  y'  sin  ^,  be  substituted 
in  equation  (1),  the  result  can  only  belong  to  points  common  to 
the  plane  and  surface,  and  will  therefore  represent  the  line  of  in- 
tersection.    Making  the  substitution  and  reducing,  we  obtain 

(wcos« (3  +  n sin2  f3)x'^  +  [cos«'J(m^i?ftfVcos8 ^)  -f  i? sin^  &]  y'^ 
+  ^m  —71)  sin  ^  cos  ^  cos  ^  a;'y'-|-  cos  (3(2ma'+  m")x' 


+  cos  ^  sin  ^{2a'm  +  m")y'-i-ma'^  +  m"a'+  I  =  0...(2). 

By  assigning  proper  values  to  a',  we  may  always  cause  the 
plane  to  intersect  the  surface,  and  by  assigning  proper  values  to 
jS  and  d,  we  may  cause  the  above  equation  to  represent  the  several 
varieties  of  lines  of  the  second  order. 


220.  For  instance,  let  it  be  required  that  the  intersection  shall 
be  a  right  line  or  lines. 

If  it  is  possible  to  cut  a  right  line  from  the  surface  by  a  plane  in 
any  position,  the  same  right  line  may  be  cut  out  by  a  plane  per- 
pendicular to  the  plane  XY.  For  it  is  only  necessary  that  the  cut- 
ting plane  should  occupy  the  position  of  the  plane  which  projects 
the  line  on  the  co-ordinate  plane  XY.  We  may  therefo'^e  regard 
6  in  the  above  equation  as  equal  to  90°,  which  gives 

cos  6  =  0,  sin  ^   =   1, 

and  see  if  it  is  possible  to  give  such  values  to  a'  and  /?,  as  will 
make  ths  equation  represent  one  or  more  right  lines. 


INDETERMINATE    GEOMETRY.  269 

221  For  those  surfaces  which  have  a  centre,  we  may  also  re- 
gard m"  =  0,  Art.  (203).  Substituting  this  value  with  the 
above,  for  cos  &  and  sin  ^,  in  equation  (2),  Art.  (219),  and  omit- 
ting the  dashes  of  x  and  y,  it  reduces  to  ' 

(m  cos'*  (3  -\-  n  sin^  (3)x'^  +  pij^  +  2a'm  cos  ^x  +  ma'^  +  /  =  0. 

Solving  this  with  reference  to  y,  we  have 

p=  dz\/—  i[(7?icos2/3+7isin2/3)a;2+2ma'cos^a;+La'«+Zj..(jlt) 

In  order  that  this  represent  one  or  more  real  right  lines,  it  is 
necessary  that     —  _     shall  be  positive,  and  that  the  factor  within 

the  parenthesis  shall  be  a  perfect  square,  Art.  (1^8),  which  re- 
quires 

P  <  0 (2), 

and 

{771  cos2  (3  +  nsm^(3){ma'^  +  I)  =  m^a'^  cos^  (3 (3). 

Deducing  the  value  of  a'  from  the  last  condition,  we  obtain 


«'=  ±\/- 


l{m  cos*  (3   +  n  sin*  ,8) 


77171  sin-" 


.(4). 


Since  p  is  positive  in  the  ellipsoid,  Art.  (209),  condition  (2)  can 
not  be  fulfilled ;  whence  the  conclusion,  that  no  right  line  can  he 
cut  from  this  surface. 

Since  m,  ti  and  /  are  posilive  in  the  hyperboloid  of  two  nappes, 
Art.  (211),  the  value  of  a'  will  be  imaginary  for  all  values  of  ^. 
Condition  (3)  can  not  then  be  fulfilled,  and  no  right  line  can  be  cut 
from  this  surface. 

Since  m  and  I  are  positive  and  n  and  p  negative  in  the  hyper- 
boloid of  one  nappe,  Art.  (213),  condition  (2)  will  be  fulfilled,  and 
the  values  of  a'  will  U  r^al  for  all  values  of  /S  which  give 


270 


INDETERMINATE    GEOMETRY. 


n  sin*  ^  <  m  cos'*  /3,         or 


n 


and  equation  (1)  will  then  represent  two  real  right  hnes  which  in*  > 
tersect,  Art.  (1*78).     Hence,  an  infinite  number  of  right  lines  may 
be  cut  from  the  surface  of  the  hyperboloid  of  one  nappe  by  planes. 


222.  If  we  take  the  value  of  Vma'^  -\-  I  from  condition 
(3)  of  the  preceding  article,  and  substitute  it  in  equation  (1),  ex- 
tract the  square  root  of  the  factor  within  the  parenthesis,  and  sub- 
stitute in  the  result  the  value  of  a',  from  equation  (4),  we  shall 
obtain 


cos  /3     /ml\ 
sin  (3  ^  pnj 


y  =  =^|      , :\/mcos^jg  -f  r^sin«;8   + 

which  may  be  put  under  the  form 

y  =   if^Xcot^^   +  n   +   cotpJ^) (1), 

and  will  represent  the   two   right  lines  cut   out  by  any  plane 
making  the  angle  ^  with  the  plane  XZ.    By  changing  (3  into  ^', 

we  shall  obtain  at  once 
the  equations  of  two  other 
lines  cut  out  by  another 
plane.  TSe  lines  cut  out 
by  two  different  planes  are 
"x  i^ot  parallel ;  for  the  cut- 
ting planes,  which  are  also 
their  projecting  planes,  are 
not  parallel.  Neither  can 
they  intersect,  for  if  they 
intersect  at  all,  it  must  be  in  the  perpendicular  to  the  plane  XY, 
at  the  point  P  ;  and  if  we  substitute  AT  for  x  in  the  equation  of 


INDETERMINATE    GEOMETRY.  27l 

the  first  set  of  lines,  and  A"P  for  x  in  the  equation  of  the  second 
set,  we  must  obtain  the  same  value  for  y  in  each  case.  But  de- 
noting the  distance  PO  by  c?,  we  have 

A'P  =  -A-  ,  A"P  =      ^ 


4 


sin  /8  sin^' 

and  these  values  being  substituted  for  a?,  each  in  equation  (1),  will 
give  values  which  are  unequal. 


223.  For  those  surfaces  which  have  no  centre,  we  may  regard 
m  and  /  as  equal  to  0,  Art.  (203).  Substituting  these  values  with 
cos  d  =  0,  sin  ^  =  1,  in  equation  (2),  Art.  (219),  and  omit- 
ting the  dashes  of  the  variables  x  and  y,  it  reduces  to 

n  sin*  ^x^  +  py*  +  m"  cos  ^x  -f  m"a'  =  0. 

Solving  this  with  reference  to  y,  we  have 


y  =   ±\/ (7isin«/3a;«  +   m"  cos  f:ix  4-  m"a') (1), 

In  order  that  this  shall  represent  one  or  more  real  right  lines, 
we  must  have,  as  in  Art.  (221), 

V  <  0 (2), 

and 

m"2  cos^  ^  =  4?i  sin*  ^  m'^a' (3)  ; 

whence 

m"  cos«  /? 


a'  = 


4»  sin*  (3 

In  the  elliptical  paraboloid,  n  and  p  are  both  positive,  Art 
(216),  condition  (2)  can  not  be  fulfilled,  and  no  right  line  can  he 
cut  from  the  surface,. 


272  INDETERMINATE    GEOMETRY. 

In  tlie  hyperbolic  paraboloid,  n  is  positive  and  p  negative,  Art. 
(218) ;  condition  (2)  is  fulfilled,  the  value  of  a'  will  be  real,  and 
fulfil  condition  (3)  for  all  values  of  ^  ;  and  an  infinite  number  of 
right  lines  may  be  cut  from  this  surface  by  planes.  Substituting 
the  value  of  a'  in  equation  (1),  and  extracting  the  square  root  of 
the  quantity  within  the  parenthesis,  we  obtain 


,      V     n     /  .     r^      ,     m"  cos 
y  =    d:  — =  (  sm  px   + 


-  p\ 


-/__  ^  \^  2  n  sin  [3  J 

which  will  represent  the  two  right  lines  cut  out  by  any  plane 
making  the  angle  (3  with  the  plane  XZ.  By  changing  /3  into  ^', 
we  shall  obtain  the  equations  of  two  other  right  lines  cut  out  by 
another  plane. 

It  may  be  proved,  as  in  the  preceding  article,  that  the  lines  cut 
out  by  two  different  planes  are  not  parallel,  and  do  not  intersect. 


224.  The  preceding  discussion  of  the  rectilineal  sections  of  sur- 
faces of  the  second  order,  enables  us  to  classify  these  surfaces  as 
they  are  classed  in  Descriptive  Geometry.     This  classification  is  : 

1.  Plane  surfaces,  which  may  be  generated  by  a  right  line 
moving  along  another  right  line  and  parallel  to  its  first  position. 

2.  Single  curved  surfaces,  which  may  be  generated  by  a  right 
line,  moving  so  that  its  consecutive  positions  shall  be  in  the  same 
plane. 

3.  Double  curved  surfaces,  which  can  only  be  generated  b}'- 
curves. 

4.  Warped  surfaces,  which  may  be  generated  by  a  right  line, 
moving  so  that  its  consecutive  positions  shall  not  be  in  the  same 
plane. 

The  cylindrical  and  conical  surfaces  are  singU  curved,  as  the  con- 
secutive elements  of  the  first  are  parallel.  Art.  (74),  and  those  of 
the  second  intersect,  Art.  (77)  ;  that  is,  are  in  the  same  plane. 


IN'OKTERAHNATE    GEOMETUr.  273 

'I'lie  ellipsoid,  hyperboloid  of  two  na|)pes,  and  elliptical  parabo- 
loid, are  double  curved ;  since  no  right  line  can  be  cut  from  them, 
Arts.  (221),  (223) ;  that  is,  no  right  line  can  be  so  placed  as  to  lie 
wholly  in  either  surface. 

The  hyperboloid  of  one  nappe,  and  hyperbolic  paraboloid,  are 
warped  ;  since  the  right  lines  cut  from  the  surfaces  by  consecutive 
planes  are  not  parallel,  neither  do  they  intersect.  Arts.  (222),  (223), 
and  therefore  can  not  lie  in  the  same  plane. 


225.  If  it  be  required  that  the  intersection  represented  by 
equation  (2),  Art.  (219),  shall  be  a  circle,  it  is  necessary  that  the 
coefficient  of  x'y'  be  equal  to  0,  and  that  the  coefficients  of  x'^  and 
y'*  be  equal  to  each  other.  Art.  (169).     This  requires 

2(7^1  —   n)  sin  /3  cos  (3  cos  &  =   0 (1), 

m  cos*  13+  n  sin*  [3  =  cos'*  &{7n  sin^  /3  +nco&'  13)  +  p  sin^  & (2). 

The  condition  (1),  (m  and  n  being  in  general  unequal),  may  oe 
satisfied  by  making  either 

sin  /3   =   0,  cos  /5   =   0,  cos  Q  =  0. 

Sin  ^   =   0     substituted  in  condition  (2),  gives 

m  —  n  cos"  &   -\-  p  sin*  ^   =  m(sin*  &   -f-  cos*  ^), 

since     sin*  &   +  cos*  ^  =   1.      From  this  we  deduce 

m  tang*  d   ■{-  ?n  =  n   ■{-  p  tang*  6, 

or 


tang^   =   ±\/IZ^. (3). 


In  a  similar  way  we  find 


CCS  13  —  0,  tang  ^  =   ^V- — Ws 

^  p  —  n 

18 


274 


INDETERMINATE    GEOMETRr. 


COS  /?     =    0, 


tang/3  =r   iV"^ i^)' 

p 


In  the  first  case,  the  cutting  plane  is  parallel  to  the  axis  of  X  ; 
in  the  second,  parallel  to  the  axis  of  Y  ;  and  in  the  third,  parallel 
to  the  axis  of  Z. 

It  may  be  remarked  that  if  any  position  of  the  cutting  plane  be 
found  to  give  a  circle,  every  parallel  plane  intereecting  the  smfac-e 
will  also  give  a  circle.  For  if  the  angles  ^  and  &  remain  the 
same,  a'  may  be  changed  at  pleasure,  without  affecting  the  equality 
of  the  coefficients  of  x'^  and  y''^. 


226.  In  the  ellipsoid,  in  which  7Ji,  n  and  p  are  positive,  Art. 
(209),  in  order  that  the  first  set  of  values  of  tang  &,  (equation 
(3),  preceding  article),  may  be  real,  we  must  have 

?i  >   m  >  p,  or  ^;  >•  m  >  n. 

In  order  that  the  second  set  may  be  real,  we  must  have 

^  >  7i  >  wi,  or  m  >  7i   >  p. 

In  order  that  the  values  of  tang  (3  may  be  real,  we  must  have 

w  >  ^  >  m,  or  ^  >  i^  >  ^• 

It  is  evident  that  no  two  of  these  conditions  can  be  fulfilled  at 
the  same  time. 

If  either  of  the  first  is  fulfilled,  we 
shall  have,  [sec  c::i-,rcssions  for  a,  h. 
and  c,  Art.  (209)], 

CE>CB>CD,  or   CE  <  CB  <CD. 


Hence,  the  mean  axis  of  the  sur 
face  lies  on  the  axis  of  X,  to  which  the  cutting  plane  is  parallel. 
If  either  of  the  second  is  fulfilled,  we  shall  have 


INDETERMINATE    GKOMETllY.  276 

CB  >  CE  >  CD  or  CB  <  CE  <  CD, 

and  for  eitlier  of  the  third 

CB  >  CD  >  CE,  or  CB  <  CD  <  CE. 

Hence,  a  cutting  plane  passed  parallel  to  the  mean  axis  of  the 
surface  may  have  two  positions,  such  that  the  sections  shall  be 
circles,  these  positions  being  determined  by  the  two  proper  values 
of  tang  &  or  tang  /3  ;  and  in  no  other  position  can  the  section 
be  a  circle. 

If  m  =  n.  both  sets  of  values  of  tang  ^  become  0,  and 
tang  /3  becomes  imaginary.  Hence  the  two  cutting  planes  unite 
in  one,  parallel  to  XY,  or  perpendicular  to  the  axis  of  Z  ;  as  should 
be  the  case,  since  the  surface  becomes  an  ellipsoid  of  revolution, 
its  axis  lying  on  the  axis  of  Z,  Art.  (210). 

If  n  =  p,  the  first  set  of  values  of  tang  &  become  imaginary, 
while  the  second  and  those  of  tang  /3  become  infinite,  and  the 
cutting  plane  is  perpendicular  to  the  axis  of  X,  Art.  (210). 

If    7)1  =  n  =  p,    the  values  of  tang  &   and  tang  ^  become  -  , 

indeterminate,  and  every  position  of  the  cutting  plane  gives  a 
circle,  as  it  should,  since  the  suiface  becomes  a  sphere. 


227.  1)1  the  hyperholoid  of  tivo  napjjes,  in  which  wi  and  n  are 
positive  and  p  negative,  the  values  of  Art.  (225),  after  giving  to 
the  letters  their  proper  signs,  become 


lang5   =   ±\/~ -,  tangd   =   ± 


m  -\-  p  ^    p  ■\'  n 


^  n  -{-  p 

ITie  values  of    tang  (3    are  imaginary. 


H^o 


^^ 


276 


INDETERMINATE    GEOMETRY. 


/O  -^   -2^ 


If  m  <^  n.  the  first  set  of  values  of  tanor 
second  imaginary, 
reverse  is  the  case. 
we  have     c  >  &  ; 


is  real,  and  the 
If  m  >  n,  the 
But,  if  m  <  w, 
and  if    w  >  n^ 


we  have  c  <  6,  Art.  (211).  Hence, 
in  this  surface,  the  cutting  plane 
must  be  parallel  to  the  longest  of 
the  two  axes  which  do  not  intersect 
the  surface. 

If    m  =  n^    the  values  of  tang  fi 

become   0,   and   the  cutting   planes 

unite  in  one  perpendicular  to  the  axis 

of  Z ;  as  they  should,  since  in  this 

case,   we  have   the   hyp^rboloid   of  revolution   of  two  nappes, 

Art.  (212). 

Since  the  above  values  of  tang  &  do  not  depend  upon  /,  they 
will  remain  the  same  when  Z  =  0,  Art.  (212),  that  is,  in  a 
cone  with  an  elliptical  base,  it  is  always  possible  to  pass  planes  in 
two  different  directions  so  as  to  cut  circles.  These  are  called  suh- 
contrary  sections.  If  one  of  them  be  regarded  as  the  base  of  the 
cone,  the  other  will  be  sub-contrary  to  the  base  ;  that  is,  in  a  scalene 
cone  with  a  circular  base,  i^  is  ahoays  possible  to  pass  a  system  of 
planes  not  parallel  to  the  base,  which  shall  cut  out  circles. 

If  the  cone  is  a  right  cone  with  a  circular  base,  it  is  a  surface 
of  revolution,  and  the  sub-contrary  sections  unite  in  one,  perpen- 
dicular to  the  axis  or  parallel  to  the  base. 


228.     In  the  hypcrholoid  of  one  nappe,  in  which  m  is  positive, 
n  and  p  negative,  we  hare 


tans  ^ 


71    +    Tfl 

,       tang  6  = 

=  ± 

V- 

n 

+ 

m 

m   +  p 

n 

P 

tang  (3  = 

V    „   _ 

m 

INDETERMINATE    GEOMETRY. 


277 


The  fii-st  are  imaginary. 

If    n  <C  p,     the  second  will  be  real  and  the  third  imaginary, 
and  the  reverse  when     n  "^  p. 
\i    n  <C  Pi     we  have 

6  =  CD  >  a  =   CA, 

and  if    n  >  p,     we  have 

CD  <  CA. 

Hence,  the  cutting  plane  is  par- 
allel to  the  greatest  of  the  two  axes 
which  pierce  the  surface. 

If  n  =  p^  the  above  real  values  of  tang  ^  and  tang  /5 
become  infinite  and  the  two  planes  unite  in  one,  perpendicular  to 
the  axis  of  X,  Art.  (214).  When  the  surface  becomes  a  cone,  the 
discussion  is  similar  to  that  in  the  preceding  article. 


229.     In  the  elli2)tical  paraboloid,  in  which     m  =   0,     n  and 
f  positive,  the  values  of    tang  &    and    tang  ^    become 


tans:  &  =  dt 


tang  ^  : 


tang  4 


V      in      /M 


p  —  n 


^   n  —  V 


The  first  are  imaginaiy.  If  n  <^  p,  the  second  are  real  and 
the  third  imaginary.  If  ^  >  JP,  the  reverse  will  be  the  case. 
Hence,  the  cutting  plane  must  be  parallel  to  the  greater  axes  of 
the  elliptical  sections.  Art.  (216). 

If  n  =  p,  the  above  real  values  become  infinite  and  the  cut- 
ting planes  unite  in  one  perpendicular  to  the  axis  of  X,  Art.  (217). 


230.     In  jthe   hyperbolic  paraboloid,   in   which     w  =   0,     n 
positive  and  p  negative,  we  have 


278  INDETERMINATE    GEOMETRY. 

tang  4    =   iv/-,  tang^    =   ±\/ !i-I, 

^  p  ^         p  -{-  n 

tang/5   =   ±\/ :^. 

The  second  and  third  are  imaginary,  and  the  first  real,  and  the 
position  of  the  cutting  plane  will  be  given  by  the  equations 


* 


sin  /3  =  0,  tang  &   =   dc  v/ • 


But  these  values  with  the  value  7n  =  0,  substituted  in  con- 
dition (2),  Art.  (225),  make  the  coefficients  of  x'^  and  y'^  both 
equal  to  0,  and  equation  (2)  of  Art.  (219),  takes  the  form 

ex  +  /  =   0,  y  indeterminate y 

which  represents  a  right  line. 

Since  any  plane  parallel  to  either  of  the  planes  determined  by 
the  above  values  of  sin  /3  and  tang  &  will  also  cut  a  right  line 
from  the  surface,  we  see  that  there  are  two  different  systems  of 
right  line  elements,  each  of  which  is  parallel  to  a  given  plane. 

Wc  conclude,  also,  that  no  circle  can  he  cut  from  the  hyperbolic 
pa^raholoid. 


231.  The  intersection  of  any  two  surfaces  of  the  second  order 
may  !•<;  found  as  in  Art.  (62) ;  but  as  their  equations  are  of  the 
fiecond  degree,  the  result  of  their  combination,  so  as  to  eliminate 
one  of  the  variables,  will  be  of  the  fourth  degree  ;  hence,  in  gene- 
ral, tlie  projections  of  the  lines  of  intersection  will  be  lines  of  the 
fourth  order,  the  discussion  of  which  will  be  complicated  and  of 
little  interest. 

If,  however,  it  is  known  that  two  such  surfaces  intersect  in  a  line 
of  the  second  order,  it  will,  in  general,  be  found  that  they  will  also 


INDETERMINATE    GEOMETRY.  27d 

intersect  in  another  line  of  the  second  order  ;  that  is,  if  one  sur- 
face enters  the  other  in  a  line  of  the  second  order,  it  will  leave  it  in  a 
line  of  the  same  order. 

To  prDve  this,  let  us  take  the  most  general  equations  of  the  two 
surfaces, 

mx^  4-  ^f^  +  pz^  +  m'xij  +  n'xz  +  i^'y^ 

+  m"x-\-  n"y   +  p"z  +  /  =   0 (1), 

qx^  +  rf/^   4-  sz^  -}-   q'xi/   -\-  r'xz  -\-   s'yz 

+  q>'x  +    r"y    +  s"z  +   l'  =   0 (2), 

and  let  the  plane  of  the  curve  in  which  it  is  known  the  two  sur- 
faces intersect  be  taken  as  the  plane  XY. 

If  we  make     z  =  0,     in  each  of  the  above  equations,  we  shall 
have 

7nx*  -f-  nf-^  +  jn'xy  -{-   m"x  -f-  ?^"y  +  ^  =   0 (3) ; 

qx''   +   ry2    -f    q'xy    -{-    q"x    +   r"y  +   Z'  =  0 (4); 

each  of  which  must  rej^resent  the  known  curve  of  intereection  of 
the  surfaces.  These  equations  must  then  be  the  same,  which  can 
only  be  the  case  when  the  corresponding  coefficients  are  equal,  or 
when  those  of  the  first  equation  are  equal  to  those  of  the  second 
multiplied  by  a  constant  factor,  as  k.  If  we  now  multiply  equation 
(2)  by  h  and  subtract  from  equation  (1),  we  obtain 

/        (p  ^  ks)z^  +  [n'  -  kr')xz  +  {p'  —  ks')yz  +  {p"  -  ks")z  =  0, 

which  equation  must  be  satisfied  for  all  values  of  x,  y  and  2,  be- 
longing to  points  common  to  the  two  surfaces.  Since  ^  is  a  com- 
mon factor,  it  may  be  satisfied  by  placing 

z  =  0, 

or 

(^  _  ks)z  +  {n'  -  kr')x  +  {p'  -  ks')y  +  (p"  —  ks'')  =  0, 


280  INDETERMINATE    GE0METR7. 

z  =  0  evidently  belongs  to  tte  plane  XY,  in  which  the  known 
line  of  intersection  lies.  The  other  is  the  equation  of  a  plane,  Art. 
(57),  which  by  its  combination  with  either  (1)  or  (2),  will  give 
another  hne  common  to  both  surfaces,  and  this  line  must,  of  course, 
be  one  of  the  second  order,  Art.  (200). 


232.  Let  x",  y"  and  z"  be  the  co-ordinates  of  a  given  point  on 
a  surface  of  the  second  order.  These  when  substituted  for  the 
variables  in  the  general  equation 

mx^  +  ny'^  +  %)Z^  ■\-  m"x  +   I  =  0 (1), 

must  satisfy  it,  and  give  the  equation  of  condition 

mx"»  +  ny"^  +  pz"^  +  m"x"   +  I  =   0 (2). 

Subtracting  this,  member  by  member,  from  equation  (1),  and 
factoring  the  terms,  we  have 

m{x  —  x")[x  4-  x")  +  '^(y  —  y'0{y  +  y") 

-{-  p(z  -   z"){z   +   z")  -\-m"{x-x")  =  0 (3), 

which  is  the  equation  of  the  surface,  with  the  condition. introduced, 
that  the  point  x",  y",  z"  shall  be  on  the  surface. 
/     The  equations  of  any  right  line  passing  through  the  given  point, 
are 

(x  -  x")  =  a(z  -  z"),  y  ^  y"  =  b{z  -  z") (4). 

If  these  equations  be  combined  with  equation  (3),  we  shall  obtain 

[z  —  z")[ma(x  +  x")  +  nh{y  +  y")  -\-p[z  +  z")  +  m"d\  =  0 (6), 

in  which  a*,  y  and  z  must  denote  the  co-ordinates  of  all  points  com- 
mon to  the  line  and  surface,  Art.  (58).  Since  this  is  an  equation 
of  the  second  degree,  there  are  but  two  such  points  ;  and  these  may 
be  determined  by  placing  the  factors  of  (5)  separately  equal  to  0. 


INDETERMINATE    GEOMETRY.  281 

z  —  z"  =  0,         gives         z  =  2",         y  =  y",         x  =  x'\, 

wliich  evidently  belong  to  the  given  point.     Placing 

ma{x  +  x")  +  nh{y  +  y'<)  -\- ]i{z  +  z")  +  m"a  =  0 (6), 

a;,  y  and  2^  in  this  must  represent  the  co-ordinates  of  the  second 
point  in  which  the  line  pierces  the  surface. 

If  now  any  plane  be  passed  through  this  right  line,  it  will  cut 
from  the  surface  a  line  which  will  contain  both  of  the  points ;  and 
if  the  second  point  be  moved  along  this  line  until  it  coincides  with 
the  first,  the  right  line  will  become  tangent  to  the  line  cut  from  the 
surface,  and  the  values  of  ic,  y  and  z  in  equation  (6),  will  become 
equal  to  x'\  y"  and  z".  Substituting  these  values  in  equation 
(6),  it  becomes 

2max"   +  2nby"  +  "Ipz"   +  m."a  =   0 (7), 

an  equation  wliich  shows  the  relation  that  must  exist  between  a 
and  h,  in  order  that  the  right  line  represented  by  equations  (4)  may 
be  tangent  to  a  line  of  the  surface  at  the  given  point ;  and  since  a 
and  h  in  this  equation  are  indeterminate,  it  follows  that  an  infinite 
number  of  right  lines  may  be  drawn,  each  tangent  to  a  line  of  the 
surface  at  the  given  point. 

If  now  in  equation  (7),  we  substitute  for  a  and  h  their  values 
taken  from  equations  (4),  we  obtain 

{2m.x"  +  m"){x  —  x")  +  '^mf'ijy  —  y")  4-  2pz"{z  —  z")  —  0, 
or,  since  from  equation  (2), 

—   2mx"^  —   2ny"^  —  Iitz"^  =   'lm"x"  -f  2/, 
we  have  finally, 

2mx''x  +  2?2y'V  +  2;V«  +  tn'\x  +  x")  +  2Z  =  0 (8), 

an  equation  which  expresses  the  relation  between  ar,  y  and  z  for  all 
points  of  the  tangent  line  in  all  of  its  positions.  The  surfoce  repre- 
sented by  it,  is  then  the  locus  of  all  right  lines  drawn  tangent  to 


282  INDETERMINATE    GEOMETRY. 

lines  of  the  surface,  at  the  given  point,  or  point  of  contact.  This 
equation  being  of  the  first  degree  between  three  variables,  is  the 
equation  of  a  plane.  This  plane  is  said  to  be  tangent  to  the  sur- 
face, at  the  given  point ;  and  in  general,  a  plane  is  tangent  to  a 
surface,  when  it  has  at  least  one  point  in  common  with  it,  through 
which  if  any  plane  he  passed,  the  sections  made  in  the  surface  and 
plane  will  he  tangent  to  each  other. 

For  those  surfaces  which  have  a  centre,  the  origin  of  co-ordinates 
being  placed  at  this  centre,  we  have  m"  =  0,  Aii.  (203),  and 
equation  (8)  reduces  to 

mxx"  -f-  nyy"   -f  P^^"   +  I  =   0 (9). 

If    m  =  n  =  j9,     equation  (9)  becomes  ^ 

xx"   -f  yy"   4-   zz"  z=   —  —  —   R^, 

m 

for  the  equation  of  a  tangent  plane  to  a  sphere,  Art.  (210). 

For  those  surfaces  which  have  no  centre,  m  =  0,  1  =  0, 
Art.  (203),  and  equation  (8)  reduces  to 

2nyy"  +  2pzz"   +  m"(x  4-  x")  =  0. 


233.  Let  x',  y'  and  z'  be  the  co-ordinates  of  a  fixed  point 
without  a  surface  of  the  second  order.  If  it  be  required  that  the 
tangent  plane  to  the  suiface  shall  pass  through  this  point,  its  co- 
ordinates must  satisfy  equation  (8),  of  the  preceding  article,  and 
give  the  equation  of  condition 

2mx"x'  +  2ny"y'  +  2pz"z'  -f  m"{x'  -f  x")  +  21  =  0... (1). 

In  this  equation  x",  y"  and  z"  are  unknown,  but  since  the  point 
which  they  represent  must  lie  on  the  surface,  we  must  also 
have  the  condition 

mx"^  +  ni/"^  4-  ;;z"=«  -f-  m"x"  +  I  =  0 (2), 


INDETERMINATE    GEOMETRY.  288 

and  these  two  equations  are  all  the  moans  which  we  have  of  de- 
termining the  values  of  x"^  y"  and  z" ;  and  since  we  thus  have 
three  unknown  quantities,  and  but  two  equations,  it  follows  that 
the  unknown  quantities  are  indeterminate.  Hence  we  conclude 
that,  in  general,  an  infinite  number  of  planes  can  be  drawn  from  a 
point  without  a  surface  of  the  second  order  tangent  to  the  surface. 
If  straight  lines  be  drawn,  from  the  different  points  of  contact  of 
these  planes,  to  the  fixed  point,  they  will  evidently  form  a  cone 
which  will  be  tangent  to  the  surface,  in  the  hue  formed  by  joining 
the  points  of  contact.  But  since  the  co-ordinates  of  these  points 
must  all  satisfy  equation  (1),  w^hen  substituted  for  x"^  y"  and  z'', 
the  points  must  lie  in  the  plane  which  will  be  represented  by  this 
equation  when  x"^  y"  and  z"  are  regarded  as  variables.  This  curve 
of  contact  must  then  be  a  plane  curve,  and  since  it  lies  on  the  sur- 
face at  the  same  time,  it  must  he  a  line  of  the  second  order,  Art. 
(200).  We  therefore  conclude  that,  in  general,  the  line  of  contact 
of  a  tangent  cone  and  surface  of  the  second  order,  is  a  line  of  the 
second  order.  And  the  same  will  be  true  of  a  tangent  cylinder,  in- 
asmuch, as  the  cone  becomes  a  cylinder,  when  its  vertex  is  re- 
moved to  an  infinite  distance. 


234.  If  it  be  required  that  the  tangent  plane  pass  through  a 
second  given  point  x'",  y'",  %'",  without  the  surface,  or  contain 
the  right  line  joining  these  two  points,  we  shall  also  have  the  equa- 
tion of  condition 

^lmx"x"'  +  2wy'y"   +  2joz'V   -f  m"{x"'  +  x")   +  2/  =   0, 

and  this  united  with  equations  (1)  and  (2)  of  the  preceding  article, 
will  give  three  equations  involving  three  unknown  quantities,  and 
since  two  of  these  equations  are  of  the  first,  and  the  other  of  the 
second  degree,  there  will  in  general  be  two  sets  of  values  for  x!\ 
y"  and  z".     Hence  we  conclude  .hat,  in  general,  two  planes  ma? 


284  INDETEEMINATE    GEOMETRT. 

be  passed  through  a  right  hne  tangent  to  a  surface  of  the  second 
order,  and  only  two. 


235.  A  right  line,  or  a  plane,  is  normal  to  a  surface  when  it  is 
perpendicular  to  a  tangent  plane,  at  the  point  of  contact. 

There  evidently  can  be  but  one  normal  line  to  a  surface  at  a 
given  point ;  but,  since  every  plane  containing  a  normal  will  be 
perpendicular  to  the  tangent  plane,  there  will  be  an  infinite  num- 
ber of  normal  planes. 


236.     The  equations  of  a  normal  line,  to  a  surface  of  the  second 
order,  will  be  of  the  form,  Art.  (50), 

X  ^  x"  =  a{x  —  z'%  y  -.  y"  =^  h(z  —  z") (1), 

in  which  it  is  necessary  to  determine  the  values  of  a  and  b  on  con- 
dition that  the  line  shall  be  perpendicular  to  the  tangent  plane 
represented  by  equation  (8),  Art.  (232).  The  equations  of  con- 
dition. Art.  (59), 


a  =   —  c, 

b 

=  -  d. 

give 

2mx"  -f  m" 
2pz"         ' 

pz" 

and  thest?,  m  equations  (1),  give 

.      ,»_2-"  +  -"(,       ,.), 

y  - 

n       ny'' 
.  v"  —    ■'  { 

2pz"  pz" 

for  the  equations  of  a  normal  line  to  any  surface  of  the  second 
order. 

By  supposing  m"  =  0,  we  shall  have  the  particular  equa- 
tions for  those  surfaces  which  have  a  centre  ;  and  by  ma]<ini:^ 
m  =  0,     we  have  them  for  those  surfaces  which  have  no  centre. 


INDETERMINATE    GEOMETRY.  285 

If    n  =  2^}     equation  (2)  reduces  to 

7JZ"     -.     7/"z    =     0, 

which,  having  no  absolute  term,  shows  that  the  projection  of  the 
normal  on  the  plane  YZ  passes  through  the  origin  of  co-ordinates ; 
Iience  the  normal  intersects  the  axis  of  X.  But  when  n  =  p^ 
the  surface  becomes  one  of  revolution,  the  axis  of  X  being  the  axis 
of  revolution,  Arts.  (210),  (214),  (21V).  We  therefore  conclude 
that  all  the  normals  to  a  surface  of  revolution  of  the  second  order 
intei'sect  the  axis  of  revolution  ;  and  that  the  meridian  plane,  pass- 
ing through  the  point  of  contact  of  a  tangent  plane,  is  a  normal 
plane  :  Or,  a  tangent  i^lctne  to  a  surface  of  revolution  of  the  second 
order  is  perpendicular  to  the  meridian  plane  passing  through  tke 
point  of  contact. 


PRACTICAL    EXAMPLES. 

237.  Although  examples  have  been  occasionally  given,  in  iin- 
mediatc  connection  with  the  articles  which  (hey  are  intended  to 
illustrate,  it  is  believed  to  be  advantageous  to  add,  in  thi'^  place,  a 
number  of  others,  a  portion  of  which  the  teacher  may  give  out 
with  each  lesson  ;  or  may  defer  them  until  the  subject  has  been 
completed,  when  their  solution  will  serve  as  a  general  review  of 
the  principles  of  the  course. 

Each  example  should  be  carefully  constructed,  on  the  black 
board,  in  proper  proportion,  a  unit  of  convenient  length  being  first 
assumed  ;  or,  when  it  can  be  done,  should  be  accurately  drawn  on 
paper,  with  mathematical  instruments.  By  this  exercise,  the 
principles  of  the  subject  will  be  strongly  impressed  upon  the 
mind  of  the  pupil,  while,  at  the  same  time,  a  good  test  of  his 
knowledge  will  be  afforded  to  his  teacher. 

The  axes  of  co-ordinates  are  supposed  to  be  at  right  angles,  un- 
less otherwise  mentioned. 


286  INDETERMINATE    GEOMETRY. 

The  teacher  may  multiply  the  examples  to  an  unlimited  extent, 
by  simply  substituting,  for  the  numbers  used,  any  others  which 
may  occur  to  him. 

1.  Construct  the  points  whose  equations  are,  Art.  (16), 

.r  =   2,  y  =   —   1  ;  .r  =   —   1,  2/  =  4  ; 

.T  =  —  3,         y=    —    2;  a:  =   3,  ?/=~5. 

2.  Find  the  expressions  for  the  distances  between  the  points, 
whose  equations  are,  Art.  (lY), 

X'  =1,  y  =   3;  X"  =   0,  y"  =    -   2; 

«'  =  -  3,  y  =   4;  a;"  =   2,  y"  =    —   1. 

3.  Construct  the  points  whose  polar  equations  are.  Art.  (18), 
V  =   20®,  r  =   5;  v  =   190°  r  =   2. 

4.  Construct  the  right  lines  whose  equations  are.  Art.  (26), 

2y  —   3x   +   1   =   0;  3y  —  a;  =   0. 

5.  Find  the  pcint  of  intersection  of  the  right  lines,  whose  equa 
tions  are  as  in  the  last  example,  Art.  (2*7). 

6.  Find  the  expression  for  the  tangent  of  the  angle,  included 
by  the  same  lines,  Art.  (28). 

Y.     Ascertain  if  the  lines  represented  by  the  equations 

2y  —   5a;    -    1   =   0,  y  =  3.i-  —   2, 

are  parallel,  or  perpendicular  to  each  other,  Art.  (28). 

8.  Find  the  equation  of  a  right  line,  passing  through  the  point 
ar'  =  2,  y'  =  —  4,  and  parallel  to  the  line  whose  equation 
is,  Art.  (30), 

3y  +  2a;   -   1   =   0. 


INDETERMINATE    GEOMETRY.  287 

9.  Find  the  equation  of  the  right  hue  passing  through  the 
same  point  and  perpendicular  to  the  same  hne,  Art.  (30) ;  also, 
the  length  of  the  perpendicular,  Art.  (l*/). 

10.  Find  the  equition  of  a  right  line  passing  through  the  two 
points, 

«'  =   3,  y  =    ^.   4;  X''  =   -   2,  y"  =   -   1. 

11.  Find  and  discuis  the  equation  of  a  circle,  the  co-ordinates 
of  whose  centre  are  a;'  =  3,  y'  ==  -—  2  ;  and  whose  radius 
is  3,  Art.  (34). 

Also,  when  the  co-ordinates  of  the  centre  are  a:  =  —  2, 
y'  =   0  ;     and  the  radius  4. 

12.  Find  the  intersection  of  the  last  circle  with  the  right  line 
whose  equation  is.  Art.  (27), 

y  =   -   ^x  -   \, 

13.  Ascertain  if  the  point  a;  =  1,  y  =  —  2,  is  on, 
without,  or  within  the  circle  whose  equation  is.  Art.  (37), 

a:8   -f    ?/8   =   9. 

14.  Find  the  equation  of  a  circle  which  shall  pass  through  the 
point     a;  =   3,     ?/  =    —   2  ;     the  origin  being  at  the  centre. 

Also  of  one  passing  through  the  point  a;  =  4,  y  =  2,  the 
origin  being  at  the  left  hand  extrsmity  of  the  diameter,  Art.  (38) 

15.  Construct  tlio  points,  in  space,  Art.  (40) ; 
x=   1,  y'=  2,  2  =  3; 
ar-=   -  2,                 y'=  3,  z"=  -  4. 

16.  Find  the  expression  for  the  distance  between  the  two  points 
given  in  last  example.  Art.  (42). 

17.  Construct  the  point  whose  polf-^r  co-ordinates  are,  Art  (43), 


288  INDETERMINATE    GEOMETRY. 

u  =   35°,  V  =   70°,  r  =   4. 

18.  Construct  the  right  line,  in  space,  whose  equations  are, 
Art.  (44), 

a;  =   2z  +   3,  y  =    -   z  +   2, 

ar.d  find  the  equation  of  its  third  projection. 

19.  Find  the  point  of  intersection  of  the  two  Jines,  in  space, 
whose  equations  are.  Art.  (47), 

a:  =   -   2z  +  3,  y  =  z  -   2  ; 

.r  =   32  — .  1,  oy  =   —   lOz  -f  2. 

20.  Find  the  expression  for  the  cosine  of  the  angle  included 
between  the  lines  given  in  the  last  example,  Art.  (49). 

21.  Ascertain  if  the  lines  whose  equations  are, 

■  a;  =   2z  +   1,  2/  =  3z  +  4 ; 

a:  =  —  2z  +    3,  y  =   z  —   2] 

are  parallel,  or  perpendicular,  Art.  (49). 

22.  Find  the  equations  of  a  right  line  which  shall  pass  through 
the  point  x'  =  —  3,  if  =  2,  z'  =  —  1,  and  be  par- 
allel to  the  line  whose  equations  are.  Art.  (49), 

a:  =    —   3z  —    1,  y  =   42  +   3. 

23.  Find  the  equations  of  a  right  line  which  shall  pass  through 
the  same  point  and  be  perpendicular  to  the  same  line,  as  in  the 
last  example.  Art.  (49). 

24.  Find  the  equations  of  a  right  line  which  shall  pass  through 
the  two  points.  Art.  (51), 

^=-1,      2/' =  2,       z'=:0;      x"=3,     y"  =  0,     2"  =  2. 


INDETERMINATE    GEOMETRY.  280 

25.  Find  the  intersection  of  tlie  two  lines  whose  equations  are, 
Art.  (53), 

0:2  4-  s*  —   5   —   0,  2+^—3   =   0; 

ar    —    3z  +   5    =   0,  z^    -{-  Ay^  —  Sy  =   0. 

26.  Find  the  equation  of  a  plane  whose  directrix  is  represent- 
ed by 

4y   —    32r  +   1    =   0, 

the  projections  of  thp  generatrix  making  angles  with  the  axis  of  Z, 
whose  tangents  are  2  and  —  3,  Art  (55). 

27.  Find  the  equations  of  the  traces  of  the  plane  represented  by 

2z  —   Sy  +  a;  +   4  =   0, 
and  the  points  in  which  it  cuts  the  co-ordinate  axes,  Art.  (56). 

28.  Find  the  point  in  which  the  right  line,  whose  equations  are 
a;  —    2z   +   2   -■=   0,  2?/   -f    3r  —    1    =   0, 

pierces  the  plane  given  in  the  last  example.  Art.  (58). 

29.  Ascertain  if  the  same  line  and  plane  are  perpendicular  to 
each  other.  Art.  (59). 

30.  Find  the  equations  of  a  right  hne  passing  through  the 
point  a;'  =  1,  y'  =  —  3,  z'  =  0,  and  perpendicular  to 
the  plane  represented  by 

'  Sx  -    4y   +   z  —    I    —   0; 

also  the  point  in  which  the  line  piercos  the  plane,  and  the  length 
of  the  perpendicular,  Art.  (60). 

31.  Find  the  expression  for  the    sine    of  the  angle  inacle  by 

the  line  whose  equations  are 
19 


290  [NDETERMINATE    GEOMETRT. 

X  =  Sz  +  5,  y  =   -  r  -f    1, 

with  the  plane  given  in  the  last  example,  Art.  (61). 

32.  Find  the  intersection  of  the  two  planes  whose  equations 
are,  Art.  (62), 

3x  —   51/  -h  z  =z   0,  X  —  t/'-3z-\-l   =   0. 

Also,  the  expression  for  the  cosine  of  the  angle  included  by  the 
«ame  planes;  Art.  (63). 

33.  Find  the  equation  of  a  plane  passing  through  the  origin 
of  co-ordinates,  and  the  two  points,  Art.  (65), 

;r'   =    -    1,  //  =   2,  2'  =   3  ; 

x"  =   0,  y"  =   —   2,  z''  =   -.  1 

34.  Tlie  equation  of  a  circle  being 

^*  +  y*  =  9; 

•find  its  equation  referred  to  a  system  of  co-ordinate  axes,  making 
an  angle  of  45°  with  each  other,  the  new  axis  of  X  being  parallel 
to  the  primitive,  and  the  new  origin  being  at  the  upper  extremity 
of  the  vertical  diameter,  Art.  (67). 

35.  Find  the  general  equation  of  the  circle  referred  to  any  set 
of  oblique  co-ordinate  axes,  Art.  (67). 

36.  Find  the  general  polar  equation  of  the  right  line.  Art.  (69). 

37.  Find  the  equation  of  a  cylinder,  the  equation  of  the  rli 
rectrix  being,  Art.  (75), 

y%  —   2x  —  a:«, 

and  the  elements  being  parallel  to  the  line, 

05  =   2s  +  4,  y  =   -   8r  4-  1. 


INDETERMINATE    GEOMETUY.  291 

38.  Find  tlie  intersection  of  the  cylinder  of  the  preceding  ex- 
ample by  the  plane  whose  equation  is,  Art.  (G2), 

3a;  —   2y  —   32;   +   2   =   0. 

39.  Find  the  general  equation  of  a  cylinder,  with  an  elliptical 
base,  the  origin  of  co-ordinates  being  at  the  centre  of  the  base, 
Art.  (15). 

40.  Find  the  equation  of  a  cone,  the  co-ordinates  of  the  vertex 
being  x'  —  1,  y'  =  2,  2'  =  —  3,  and  the  equation  of 
the  directrix,  Art.  (V7), 

?/«  =   Gx. 

41.  Find  the  intersection  of  the  same  cone  by  the  plane  whose 
equation  is,  Art.  (62), 

X  +  21/   ~   Sz  =  0. 

42.  Find  the  equation  of  a  right  cone,  the  equation  of  the  di- 
rectrix being 

x^  +  ,/  =   9,  ^  (J 

♦he  altitude  being  5,  Art.  (77). 


43.  Intersect  the  same  cone  by  a  plane  passing  through  the 
axis  of  Y  and  making  an  angle  of  45°  with  the  base,  and  find  the 
equation  of  the  intersection  in  its  own  plane,  Art.  (81). 

44.  Find  the  general  equation  of  a  cone  with  a  hyperbolic  base, 
the  origin  of  co-ordinates  being  at  the  centre  of  the  base,  Art.  (77). 

45.  Construct  the  parabolas  whose  equations  are,  Arts.  (85), 
(86), 

yi  =   4x;  y«  =   -   3a:;  .r«  =  9y. 

40.     Ascertain  whether  the  point    a;'  =    —  3,     y'  =   3,     i« 


292  INDETERMINATE    GEOMETRY, 

without,  on,  or  within  each  of  the  parabolas  given  in  the  preceding 
example.  Art.  (87). 

47.  Find  the  equation  of  a  parabola  which  shall  pass  through 
the  point     x'  =   3,      y'  —   5,    Art.  (29). 

48.  Find  the  intersection  of  the  circle  and  parabola  whoso 
equations  are,  Art.  (27), 

a;«  +  y«   =   6,  y^  —   2x, 

49.  Find  the  equation  of  a  tangent  to  the  parabola  y^  =  —  2.i, 
at  the  point     y'  =  4,     x'  =    —  8,  Art.  (90). 

Find  also  the  equation  of  a  normal  at  the  same  point.  Art.  (98). 

50.  Find  the  equation  of  a  tangent  to  the  parabola  y*  =:  4.r, 
and  parallel  to  the  right  line  whose  equation  is,  x\rts.  (30),  (OO). 

2y  =   3x   +   5. 

51.  Find  the  equations  of  the  two  tangents  to  the  parabola 
represented  by  y^  =  6.r,  which  shall  pass  through  the  point 
X'  =   1,     y'  =   4,     Art.  (93). 

52.  Find  the  equation  of  the  polar  line  to  the  point  c  =  2, 
rf  =  1,     for  the  parabola  represented  by     y'  =   3.r,     Art.  (95). 

53.  The  equation  of  the  polar  line  to  the  same  parabola  beinor 

y  ==  .-r  +   2, 
find  the  co-ordinates  of  the  polo,  Art.  (95), 

54.  The  equation  of  a  parabola  being  y^  =  4.r,  find  its 
equation  when  referred  to  a  diameter  and  tangent  at  its  vertex, 
the  tangent  making  an  angle  of  45°  with  the  axis,  Art.  (99). 

55.  Determine  the-  axes,  and  construct  the  ellipses,  whose 
equations  are.  Art.  (106),  ' 

2y«  -f  3^«  =   4-  4y^   +  a.'2  =  9. 


INDETERMINATE    GEOMETRY.  293 

56.  Determine  the  axes  and  construct  tlie  hyperbolas,  whose 
equations  are,  Art.  (10 7), 

2/2  —  3a;2  =    —   5;  2?/8  —  4a;«  =   4. 

57.  Ascertain  whether  the  point  x'  =2,  y'  =  3,  is 
without,  on,  or  within  each  of  the  curves  given  in  the  last  two  ex- 
amples, Arts.  (109),  (110). 

58.  Find  the  equation  of  an  ellipse  which  shall  pass  through 
the  point  x'  =  3,  y'  =  2,  the  origin  of  co-ordinates  being  at 
the  centre,  and  the  semi-transverse  axis  equal  to  4,  Art.  (125). 

59.  Find  the  equation  of  an  hyperbola  which  shall  pasy 
through  the  point  a;'  =  —  3,  y'  =  —  2,  the  origin  being 
at  the  centre,  and  the  semi-conjugate  axis  equal  to  2,  Art.  (126). 

60.  'Find  the  intersection  of  the  ellipse  and  parabola,  whose 
equations  are.  Art.  (27), 

2y2  4.  43.2  —   8,  ys  ==   _   5^^ 

61.  Find  the  intersection  of  the  ellipse  and  hyperbola,  whose 
equations  are,  Art.  (27), 

37/2   ^   ^.a  —   3^  23/2   _   3a;S   =    -    6. 

62.  Find  the  equations  of  a  tangent  and  normal,  to  the  ellipse 
represented  by 

4y2  4-  a;«  =   9, 
at  the  point     x"  =   1 ;      y"  =  '/2,     Art.  (128). 

63.  Find  the  equations  of  a  tangent  and  normal  to  the  hyper- 
bola 

4!/2  —   2x^  —   —  8, 
at  the  point     .r"  =  Vs,     y"  =    V2,     Arts.  (131),  (30). 


294  INDETERMINATE    GEOMETKr. 

64.     Find  the  eqimticn  of  a  tangent  to  tlie  ellipse 
4y«  +   9.r«  =   36, 
and  making  the  angle  45°  with  the  axis,  Arts.  (30),  (128). 

05.  Find  the  equations  of  the  two  tangents  to  the  ellipse  re- 
J^resented  by 

4y2  4-  3x^  =   12, 
which  shall  pass  through  the  point    x   =  1,     y  =  4,    Art.  (133). 

66.  Find  the  equations  of  the  two  tangents  to  the  hyperbola 
represented  by 

y2  —   3.r8  =   —   5, 
which  shall  pass  through  the  point    x'  =  2,     y'  =  3,    Art.  (134). 

67.  The  equations  of  an  ellipse  and  its  polar  line  being 

42/2  +  2a;2  =  8  ;  y  =  2x  -\-  6, 

find  the  co-ordinates  of  the  pole,  Art.  (139). 

68.  The  equation  of  an  hyperbola  being 

3y3  _   2x^  r=    —   6, 

find  tlie  equation  of  the  polar  line  of  the  pole     c  =  4,     d  =  0^ 
Art.  (140). 

69.  Construct  an  ellipse,  the  two  conjugate  diameters  of  which 
are  6  and  4,  making  an  angle  of  120°  ;  also  an  hyperbola  having 
the  same  conjugate  diameters,  Art.  (150). 

10.     Find  the  position  and  length  of  the  equal  conjugate  di 
ainetera  of  the  ellipse,  whose  equation  is,  Art.  (159), 

4y2  4-  Sx^  =   12. 
Yl.     Construct  the  asymptotes  of  the  hyperbola, 


INDETERMINATE    GEOMETRY.  296 

4y«  —  2a:«  =   —   8, 

and  find  its  equation  when  referred  to  them.  Art.  (161). 

72.     Construct  the  hyperbola  whose  equation  is,  Art.  (170), 

2X1/  -\-  Sij  +  X  —   1   =  0, 

T3.  For  examples  illustrating  the  discussion  of  the  general 
equation  of  the  second  degree  between  two  variables,  see  Arts. 
(173),  (176),  (179). 

74.  Ascertain  if  the  line  represented  by  the  equation 

y*  ^   x^  —   2x  —   3   =   Oj 
has  a  centre,  and  determine  its  co-ordinates.  Art.  (181). 

75.  For  examples  relating  to  loci,  see  Art.  (194). 

76.  Find  the  equation  of  the  surface  generated  by  revolving 
the  right  line  whose  equations  are 

4x  =   3s  -f   2,  2y  =   —  «  +  6, 

about  the  axis  of  Z,  Art.  (196). 

77.  Find  the  equation  of  the  paraboloid  of  revolution  generated 
by  the  parabola  represented  by,  Art.  (198), 

y2   =    _    3x. 

78.  Find  the  equations  of  the  spheroids,  generated  by  tlw 
ellipse  represented  by 

4y«   4-   a:2   =   4. 

79.  Find  the  equations  of  the  hyperboloids,  generated  by  the 
hyperbola  represented  by 

92/8   _   4a:«  =    —   3C. 


296  INDETERMINATE    GEOMETRY 

80.     Find  the  equation  of  the  surface  generated  by  revolving 
the  parabola  represented  by 


about  the  axis  of  Y. 

81.  Find  the  equations  of  the  surfaces  generated  by  revolving 
the  lines  represented  by 

2/«  =  i,  f  =  2x\ 

X 

nbout  the  axis  of  Y. 

Also  the  surface  generated  by  revolving  the  first  hne  about  the 
axis  of  X. 

82.  Find  the  position  of  the  planes  which  will  make  circular 
sections,  Art.  (226),  in  the  ellipsoid  whose  equation  is 

2x^  4-   3y2  +  4z»  =   1. 

83.  Find  the  position  of  the  planes  which  will  make  circular 
sections,  Arts.  (227),  (228),  in  the  hyperboloids  whose  equations 
are 

x«   +   2?/2   —   2*   =    —    3  ;  4a:«   —   y*   -    32«   =    —    2. 

84.  Find  the  position  of  the  planes  which  will  make  circular 
sections,  Art.  (229),  in  the  paraboloid  whose  equation  is 

2y«   +   32«  _  4^  =   0. 

85.  P'ind  the  equation  of  a  tangent  plane,  Art  (232),  to  the 
ellipsoid,  whose  equation  is 

4x'  4-   2y«  -f  «•  =   10, 

at  the  point  whose  co-ordinates  are       x"  =   1,      y"  =   —  1, 
j"  =   2. 


INDETERMINATE    GEOMETRY.  297 

Also  the  equation  of  a  normal   line,  Art.  (230),  at  the  same 
point. 

80.     Find  the  equations  of  the  tangent  planes,  Art.  (232),  and 
normal  lines  Art.  (230),  to  the  hyperboloids  whose  equations  are 

2;r8  -f  y2  _    ovi  ^  _    X8  ;  l]x^  —   2y^  —   z^   =  —  1  •, 

at  the  point  of  the  first,  represented  by  x"  =  2,  ;y"  =  —  l , 
z"  =  3  ;  and  at  the  point  of  the  second,  represented  by 
x'   =   2,     y"   =    -   3  ;     ::"   =    1. 

87.     Find  the  equations  of  the  tangent  planes.  Art.  (232),  and 
normal  lines.  Art.  (236),  to  the  paraboloids  whose  equations  are 

2.?/«  -f   32«  =   4x ;  4//2  —   z^  —   5x ; 


at  the  jwint  of  the  first,  represented  by  x"  =  o,  y"  =  2, 
z"  =  L-  2 ;  and  at  the  point  of  the  second,  represented  by 
jt'    =^,      y"  =    -   3,      z"  =   -   4. 


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